LIBRARY OF CONGRESS. 



CliapXJ.l&.^^opy right No. 

Shelf..:.a33 

UNITED STATES OF AMERICA. 



FORMULAS 



IN 



GEARING 






I' 



33 



WITH PRACTICAL SUGGESTIONS. 



Ui, 



>?n«1'^'' 



PROVIDENCE, R. I. 

BROWN & SHARPE MANUFACTURING COMPANY, 

1896. 



"x •■■■ 



Entered according to Act of Congress, in the year 1896 by 

BROWN & SHARPE MFG. CO., 

In the Office of the Librarian of Congress at "Washington. 

Registered at Stationers' Hall, London, Eng. 

All rights reserved. 



.v1 



Ofo -3f Ss"? 



PREFACE 



It is the aim, in the following pages, to condense as much 
as possible the solution of all problems in gearing which in the 
ordinary practice may be met with, to the exclusion of prob- 
lems dealing with transmission of power and strength of 
gearing. The simplest and briefest being the symbolical 
expression, it has, whenever available, been resorted to. The 
mathematics employed are of a simple kind, and will present 
no difficulty to anyone familiar with ordinary Algebra and 
the elements of Trigonometry. . ' 



CONTENTS, 

FORMULAS IN GEARING. 



CHAPTER I. 

Page 
Systems of Gearing . . . . i 

CHAPTER n. 

Spur Gearing — Formulas — Table of Tooth Parts — Comparative Sizes 

of Gear Teeth 4. 

CHAPTER HI. 

Bevel Gears, Axes at Right Angles — Formulas — Bevel Gears, Axes at 
any Angle — Formulas — Undercut in Bevel Gears — Diameter Incre- 
ment — Tables for Angles of Edge and Angles of Face — Tables of 
Natural Lines 11 

CHAPTER IV. 

Worm and Worm Wheel, Formulas — Undercut in Worm Wheels — 

Table for gashing' Worm Wheels 34 

CHAPTER V. 

Spiral or Screw Gearing — Axes Parallel — Axes at Right Angles — 
Axes at any Angle — General Formulas— Table of Prime Num- 
bers and Factors 39 

CHAPTER VI. 

Internal Gearing — Internal Spur Gearing — Internal Bevel Gears cy 

CHAPTER VII. 
Gear Patterns 6^ 

CHAPTER VIII. 
Dimensions and Form for Bevel Gear Cutters 66 

CHAPTER IX. 
Directions for cutting Bevel Gears with Rotary Cutter 69 

CHAPTER X. 

The Indexing of any Whole or Fractional Number 72 

CHAPTER XI. 

The Gearing of Lathes for Screw Cutting — Simple Gearing — Compound 

Gearing — Cutting a Multiple Screw 76 



FORMULAS IN GEARING. 



Ch:af»te;r I. 



SYSTEMS OF GEARING. 

(Figs. I, 2.) 

There are in common use two systems of gearing, viz.: the 
involute and the epicycloidal. 

In the involute system the-outlines of the working parts of a 
tooth are single curves, which may be traced by a point in a 
flexible, inextensible cord being unwound from a circular disk 
the circumference of which is called the base circle^ the disk 
being concentric with the pitch circle of the gear. 




Fl(j, 1. 



In Fig. I the two base circles are represented as tangent to 
the line P P. This line (P P) is variously called " the line of 
pressure," " the line of contact," or '' the line of action." 



2 BROWN & SHARPE MFG. CO. 

In our practice this is drawn so as to make with a normal 
to the center line (O O') 14%°, or with the center line 75}^°. 

The rack of this system has teeth with straight sides, the two 
sides of a tooth making, together, an angle of 29° (twice 

14)^°). 

This applies to gears having 30 teeth or more. For gears 
having less than 30 teeth special rules are followed, which are 
explained in our " Practical Treatise on Gearing." 




Fig, 2, 



In epicycloidal^ or double-curve teeth, the formation of the 
curve changes at the pitch circle. The outline of the faces of 
epicycloidal teeth may be traced by a point in a circle rolling 
on the outside of pitch circle of a gear, and the flanks by a point 
in a circle rolling on the inside of the pitch circle. The faces 
of one gear must be traced by the same circle that traces the 
flanks of the engaging gear. 

In our practice the diameter of the rolling or describing 
circle is equal to the radius of a 15-tooth gear of the pitch 
required ; this is the base of the system. The same describing 
circle being used for all gears of the same pitch. 



The teeth of the rack of this system have double curves, 
which may be traced by the base circle rolling alternately on 
each side of the pitch line. 

An advantage of the involute over the epicycloidal tooth is, 
that in action gears having involute teeth may be separated a 
little from their normal positions without interfering with the 
angular velocity, which is not possible in any other kind of 
tooth. 

The obliquity of action is sometimes urged as an objection 
to involute teeth, but a full consideration of the subject will 
show that the importance of this has been greatly over-esti- 
mated. 

The tooth dimensions for both the involute and epicycloidal 
gears may be calculated from the formulas in Chapter II. 



BROWN & SHARPE MFG. CO. 



CHAPTKR II. 

SPUR GEARING. 

(Figs. 3, 4.) 

Two spur gears in action are comparable to two correspond- 
ing plain rollers whose surfaces are in contact, these surfaces 
representing the pitch circles of the gears. 

Pitch of Gears. 

For convenience of expression the pitch of gears may be 
stated as follows : 

Circular pitch is the distance from the center of one tooth to 
the center of the next tooth, measured on the pitch line. 

Diametral pitch is the number of teeth in a gear per inch of 
pitch diameter. That is, a gear that has, say, six teeth for each 
inch in pitch diameter is six diameiial pitch, or, as the expres- 
sion is universally abbreviated, it is " six pitch." This is by 
far the most convenient way of expressing the relation of 
diameter to number of teeth. 

Chordal pitch is a term but little employed. It is the dis- 
tance from center to center of two adjacent teeth measured in 
a straight line. 




PROVIDENCE, R. I. 



FORMULAS. 



N = number of teeth. 

s = addendum. 

t = thickness of tooth on pitch line. 

/= clearance at bottom of tooth. 
D" = working depth of tooth. 
D" + / = whole depth of tocch. 

d= pitch diameter. 
d' = outside diameter. 
P' = circular pitch. 
P^ = chord pitch. 

P = diametral pitch, 
C = center distance. 



p_ N + 2 

p z= -ZL 
P' 

p' = 5 
p 



f=^ = .3x83P' 



__d ^ 



N N -f 2 

2 2 P 



lO 

i8o° 



lO 

s 

D" = 2S 

P^ — . ^sin 



N 
360° d 



d P" 

P' = dTt where sin S = — 



d = .^ 

d' = d -h 2 s 

. NP' 



BROWN & SHARPE MFG. CO. 

GEAR WHEELS. 

TABLE OF TOOTH PARTS CIRCULAR PITCH IN FIRST COLUMN. 



p' 

2 


Threads or 

Teeth per inch 

Linear. 


1 . 
11 


Thickness of 

Tooth on 
Pitch Line. 




h 

r 


Depth of Space 

below 

Pitch Line. 


OH 


Width of 

Thread-Tool 

at End. 




f 


P 


t 


s 


D" 


s+f 


1.3732 


P'x.31 


P'x.335 


i 


1.5708 


1.0000 


.6366 


1.2732 


.7366 


.6200 


.6700 


1^ 


A 


1.6755 


.9375 


.5968 


1.1937 


.6906 


1.2874 


.5813 


.6281 


If 


i 


1.7952 


.8750 


.5570 


1.1141 


.6445 


1.2016 


.5425 


.5863 


If 


A 


1.9333 


.8125 


.5173 


1.0345 


.5985 


1.1158 


.5038 


.5444 


li 


1 


2.0944 


.7500 


.4775 


.9549 


.5525 


1.0299 


.4650 


.5025 


h\ 


a 


2.1855 


.7187 


.4576 


.9151 


.5294 


.9870 


.4456 


.4816 


^ 


t\ 


2.2848 


.6875 


.4377 


.8754 


.5064 


.9441 


.4262 


.4606 


lA 


ii 


2.3936 


.6562 


.4178 


.8356 


.4834 


.9012 


.4069 


.4397 


li 


i 


2.5133 


.6250 


.3979 


.7958 


.4604 


.8583 


.3875 


.4188 


ItV 


n 


2.6456 


.5937 


.3780 


.7560 


.4374 


.8156 


.3681 


.3978 


14 


f 


2.7925 


.5625 


.3581 


.7162 


.4143 


.7724 


.3488 


3769 


lA 


H 


2.9568 


.5312 


.3382 


.6764 


.3913 


.7295 


.3294 


.3559 


1 


1 


3.1416 


.5000 


.3183 


.6366 


.3683 


.6866 


.3100 


.3350 


n 


lA 


3.3510 


.4687 


.2984 


.5968 


.3453 


.6437 


.2906 


.3141 


1 


H 


3.5904 


.4375 


.2785 


.5570 


.3223 


.6007 


.2713 


.2931 


a 


1^ 


3.8666 


.4062 


.2586 


.5173 


.2993 


.5579 


.2519 


.2722 


f 


1* 


4.1888 


.3750 


.2387 


.4775 


.2762 


.5150 


.2325 


.2513 


n 


lA 


4.5696 


.3437 


.2189 


.4377 


.2532 


.4720 


.2131 


.2303 


i 


li 


4.7124 


.3333 


.2122 


.4244 


.2455 


.4577 


.2066 


.2233 



PROVIDENCE, R. L 
TABLE OF TOOTH TARTS.— Continued. 

CIRCULAR PITCH IN FIRST COLUMN. 



a 
55 


Threads or 

Teeth per inch 

Linear. 


Diametral 
Pitch. 


Thickness of 

Tooth on 
Pitch Line. 


-k 

id 

< 

s 


M 1 Working Depth 
^ 1 of Tooth. 


Depth of Space 

beloAV 

Pitch Line. 


Whole Depth 
of Tooth. 


Width of 

Thread-Tool 

at End. 


Width of 
Thread at Top. 


p' 


1" 


P 


t 


^^/ 


D"+/. 


P'x.31 


P'x.835 


1} 


5.0265 


.3125 


.1989 


.3979 


.2301 


.4291 


.1938 


.2094 


A 


n 


5.5851 


.2812 


.1790 


.3581 


.2071 


.3862 


.1744 


.1884 


i 


2 


6.2832 


.2500 


.1592 


.3183 


.1842 


.3433 


.1550 


.1675 


iV 


2* 


7.1808 


.2187 


.1393 


.2785 


.1611 


.3003 


.1356 


.1466 


1 


2i 


7.8540 


.2000 


.1273 


.2546 


.1473 


.2746 


.1240 


.1340 


f 


2| 


8.3776 


.1875 


.1194 


.2387 


.1381 


.2575 


.1163 


.1256 


4 


3 


9.4248 


.1666 


.1061 


.2122 


.1228 


.2289 


.1033 


.1117 


iV 


H 


10.0531 


.1562 


.0995 


.1989 


.1151 


.2146 


.0969 


.1047 


.2 


3} 


10.9956 


.1429 


.0909 


.1819 


.1052 


.1962 


.0886 


.0957 


i 


4 


12.5664 


.1250 


.0796 


.1591 


.0921 


.1716 


.0775 


.0838 


2 

y 


4i 


14.1372 


.1111 


.0707 


.1415 


.0818 


.1526 


.0689 


.0744 


i 


5 


15.7080 


.1000 


.0637 


.1273 


.0737 


.1373 


.0620 


.0670 


3 


GJ 


16.7552 


.0937 


.0597 


.1194 


.0690 


.1287 


.0581 


.0628 


J, 
6 


6 


18.8496 


.0833 


.0531 


.1061 


.0614 


.1144 


.0517 


.0558 


i 


7 


21.9911 


.0714 


.0455 


.0910 


.0526 


.0981 


.0443 


.0479 


J 


8 


25.1327 


.0625 


.0398 


.0796 


.0460 


.0858 


.0388 


.0419 


1 


9 


28.2743 


.0555 


.0354 


.0707 


.0409 


.0763 


.0344 


.0372 


I'o 


10 


31.4159 


.0500 .0318 


.0637 


.0368 


.0687 


.0310 


.0335 


1 . 

1 C 


IG 


50.2655 


.0312 .0199 


.0398 


.0230 


.0429 


.0194 


.0209 



BROWN & SHARPE MFG. CO. 



GEAR WHEELS. 



TABLE OF TOOTH PAKTS DIAMETKAL PITCH IN FIRST COLUMN. 



1 

Diametral 
Pitch. 




Thickness 
of Tooth on 
i Pitch Line. 




II 

o ° 


Depth of Space 

below 

Pitch Line. 


Whole Depth 
of Tooth. 


P 


P' 


t 


s 


D" 


s+f. 


D"+/. 


i 


6.2832 


3.1416 


2.0000 


4.0000 


2.3142 


4.3142 


f 


4.1888 


2.0944 


1.3333 


2.6666 


1.5428 


2.8761 


1 


3.1416 


1.5708 


1.0000 


2.0000 


1.1571 


2.1571 


li 


2.5133 


1.2566 


.8000 


1.6000 


.9257 


1.7257 


li 


2.0944 


1.0472 


.6666 


1.3333 


.7714 


1.4381 


If 


1.7952 


.8976 


.5714 


1.1429 


.6612 


1.2326 


2 


1.5708 


.7854 


.5000 


1.0000 


.5785 


1.0785 


2i 


1.3963 


.6981 


.4444 


.8888 


.5143 


.9587 


2i 


1.2566 


.6283 


.4000 


.8000 


.4628 


.8628 


2J 


1.1424 


.5712 


.3636 


.7273 


.4208 


.7844 


3 


1.0472 


.5236 


.3333 


.6666 


.3857 


.7190 


3i 


.8976 


.4488 


.2857 


.5714 


.3306 


.6163 


4 


.7854 


.3927 


.2500 


.5000 


.2893 


.5393 


5 


.6283 


.3142 


.2000 


.4000 


.2314 


.4314 


6 


.5236 


.2618 


.1666 


.3333 


.1928 


.3595 


7 


.4488 


.2244 


.1429 


.2857 


.1653 


.3081 


8 


.3927 


.1963 


.1250 


.2500 


.1446 


.2696 


9 


.3491 


.1745 


.1111 


.2222 


.1286 


.2397 


10 


.3142 


.1571 


.1000 


.2000 


.1157 


.2157 


11 


.2856 


.1428 


.0909 


.1818 


.1052 


.1961 


12 


.2618 


.1309 


.0833 


.1666 


.0964 


.1798 


13 


.2417 


.1208 


.0769 


.1538 


.0890 


.1659 


14 


.2244 


.1122 


.0714 


.1429 


.0826 


.1541 



PROVIDENCE, R. I. 



TABLE OF TOOTH TA'RTS— Continued. 



DIAMETRAL PITCH IN FIRST COLUMN. 



■g . 

II 
r 


5^ 


Thickness 
of Tooth on 
Pitch Line. 


IS 


Working Deptli 
of Tooth. 


Depth of Space 

below 

Pitch Line. 




p. 


P'. 


t. 


s. 


D". 


.0771 


D"-^/. 


15 


.2094 


.1047 


.0666 


.1333 


.1438 


16 


.1963 


.0982 


.0625 


.1250 


.0723 


.1348 


17 


.1848 


.0924 


.0588 


.1176 


.0681 


.1269 


18 


.1745 


.0873 


.0555 


.1111 


.0643 


.1198 


19 


.1653 


.0827 


.0526 


.1053 


.0609 


.1135 


20 


.1571 


.0785 


.0500 


.1000 


.0579 


.1079 


22 


.1428 


.0714 


.0455 


.0909 


.0526 


.0980 


24 


.1309 


.0654 


.0417 


.0833 


.0482 


.0898 


26 


.1208 


.0604 


.0385 


.0769 


.0445 


.0829 


28 


.1122 


.0561 


.0357 


.0714 


.0413 


.0770 


30 


.1047 


.0524 


.0333 


.0666 


.0386 


.0719 


32 


.0982 


.0491 


.0312 


.0625 


.0362 


.0674 


34 


.0924 


.0462 


.0294 


.0588 


.0340 


.0634 


36 


.0873 


.0436 


.0278 


.0555 


.0321 


.0599 


38 


.0827 


.0413 


.0263 


.0526 


.0304 


.0568 


40 


.0785 


.0393 


.0250 


.0500 


.0289 


.0539 


42 


.0748 


.0374 


.0238 


.0476 


.0275 


.0514 


44 


.0714 


.0357 


.0227 


.0455 


.0263 


.0490 


46 


.0683 


.0341 


.0217 


.0435 


.0252 


.0469 


48 


.0654 


.0327 


.0208 


.0417 


.0241 


.0449 


50 


.0628 


.0314 


.0200 


.0400 


.0231 


.0431 


56 


.0561 


.0280 


.0178 


.0357 


.0207 


.0385 


60 


.0524 


.0262 


.0166 


.0333 


.0193 


.0360 



lO 



BROWN & SHARPE MFG. CO. 



Comparative Sizes of Gear Teeth. 
Involute. 




8 p 



Fiij* 4, 



PROVIDENCE, R. I. 



I I 



CHAPTER III. 

BEVEL GEARS.— AXES AT RIGHT ANGLES. 

(Fig. 5.) 




12 BROWN & SHARPE MFG. CO. 



FORMULAS. 

5^« ^ I Number of teeth \ ^^^r* 
N5 = j ( pinion 

P = diametral pitch. 

P' = circular pitch. 

a^ = \ center angle = angle of edge j gear. 
a^j =z ) or pitch angle ( pinion. 

/3 = angle of top. 

/?' = angle of bottom. 

^« = I angle of face \ ^^^^' 
gb= \ ^ I pmion. 

j"'~ !- cutting angle K ^. .' 
/lb = ) & & ^ pmion. 

A = apex distance from pitch circle. 

A' = apex distance from large bottom of tooth. 

d = pitch diameter. 
^' = outside diameter. 

s = addendum. 

/ = thickness of tooth at pitch line. 
/ = clearance at bottom of tooth. 
D" = working depth of tooth. 
D" + /= whole depth of tooth. 
2 a = diameter increment. 

/^ = distance from top of tooth to plane of pitch circle. 
F = Vi^idth of face. 



PROVIDENCE, R. I. 13 



N N, 

tan a„ = — — - — : tan cxj 



^ 2 sin a 4. z? i" 

tan p = — — — ; or tan p = — • 

t^^fy = ^'^no^(^ + Td ^^-3^4sina tan // = £±/ ; 

N N ^ ■ A ' 

h = a— ff [See Note, pages 2.) 

2 



'^'A'^'^m 



A= N 



2 P sin a 



A'= _^ A' = N 



COS p' 2 P sin a cos /:^' 

A = i^ cos ^ 

sin {a + /^) 

p= N 



2 A sin a 



V N N P' .... 

a = — or = a = a -\- 2 a 

P Tt 

2 a =^ 2 s cos a (Seepage 20.) 

b = ais.na i'^f°'-g^a.>- =f for pinion 
( a for pinion = for gear 

P = —- P'= i^ 

P' P 

s = 1 z=z^= .3183 P' ^ = Atan/? 

P TT ^ ^ ^ 

j- + / = . 3685 P' ^ + / = A tan /5' 

2 2 P 

F = i + ^- or=: 2P' to3P' 
P 7 



Note. —Formulas containinj^ notations without the designating: letters a and b 
apply equally to either g:(ar or pinion. If wanted for one or the other, the respective 
letters are simply attached. 



BROWN v«t bHAKPE MFG. CO. 



BEVEL GEARS WITH AXES AT ANY ANGLE. 




Fifi. a. 



PROVIDENCE, R. 1. 15 



FORMULAS. 

C = angle formed by axes of gears. 

t^t" ~ !■ number of teeth -] ^. . ' 
N5 = j ( pinion. 

P = diametral pitch, 

P' = circular" pitch. 

" Z f angle of edge = pitch angle -j ^9^^- 

fd = angle of top. 
fi' = angle of bottom. 



I angle of face -I ^?^}'' 
] ^ { pinio 

/' ~ >- cutting: ansfle < ^. . 
/lb = ) & & ( pinic 



ion. 

A — apex distance from pitch circle. 
A' = apex distance from large bottom of tooth. 

d = pitch diameter. 
d' = outside diameter. 
2a = diameter increment. 

d = distance from top of tooth to plane of pitch circle. 



Note. — The formulas for tooth parts as given on page 5 apply equally to thee 
cases. 



l6 BROWN & SHARPE MFG. CO, 



sin C , Nft , , ^ 

tan a^— — ; or cot a^ = — — A_^ + cot C 

1^^-hcosC ^«'^^^ 

tan a^ = ^^ ^'"^ ^ ; or cot a^ = ^^ "^^ ^ + cot C 

N„ , ., Nft 3in C 

— 2: + cos S^ ^ 

Note. — These formulas are correct only for values of C less than 90°. If C 
greater than 90°, consult the following: page. 



^ 2 sm a ^ JD s 

tan p = — - — ; or tan /? = _ ; 

N A 

ga=9o" - K+ /?) ; g^= 90° - («'5 + /?) 

h = a — p {See Note, page ^2.) 

N 



A 


2 


i-N 






P sin or 


A' 


A 

cos/5' 




^/ = 


N 
" P 


or = 


NF 

7t 



d' •=zd -\- 2 a 



2 a =^ 2 s cos a 

a for gear z=z b for pinion. 
a for pinion ■= b for gear. 

Note. — See Foot Note on page 13. 



PROVIDENCE, R. I. 



17 




l8 BROWN & SHARPE MFG. CO. 



The formulas given for a^^ and a^ (when C, N^ and N^ are 
known) undergo some modifications for values of C greater 
than 90°. 

For bevel gears at any angle but 90° we may distinguish 
four cases ; C, N^, N^ being given. 

/. Case. See pages 14 and 16. 

//. Case. C is greater than 90°. 
tan «„= sin (180 -C) ^^^ ^ ^\r^ (iZo - C 



^-cos(i8o-C) — "-cos(i8o-C; 



tan o'j, 

N, -' N 

///. Case. (Xa — 90° ; ^'^ = C — 90° 
IV. Case. 

sin E ^ sin E 

tan (Xa = ; tan a^, = 



COS E — - -'' _" — cos E 



N. N 



For an example to apply to Case III., the following condi- 
tion must be fulfilled : 

N„ sin (C - 90°) = N, 

To distinguish whether a given example belongs to Case II. 
or case IV., we are guided by the following condition : 

Is • N sin (C - 00°) \ ^^^^^^^ ^^^^ ^b, we have Case II. 
" ^ ( l<^i"ger than Nft, we have case IV. 



PROVIDENCE, R. I. I9 



UNDERCUT IN BEVEL GEARS. 

By undercut in gears is understood a special formation of 
the tooth, which may be explained by saying that the elements 
of the tooth below the pitch line are nearer the center line of 
the tooth than those on the pitch line. Such a tooth outline is 
to be found only in gears with few teeth. In a pair of bevel 
gears where the pinion is low-numbered and the ratio high, we 
are apt to have undercut. For a pair of running gears this 
condition presents no objection. Should, however, these gears 
be intended as patterns to cast from, they would be found use- 
less, from the fact that tliey would not draw out of the sand. 
We have stated on page 2 (see Fig. i) that the base of our 
involute system is the 14^° pressure angle. If a pair of bevel 
gears with teeth constructed on this basis have undercut, we 
can nearly eliminate the undercut — and for the practical work- 
ing this is quite sufficient— by taking as a basis for the con- 
struction of the tooth outline a pressure angle of 20°. 

The question now is : When do we, and when do we not 
have undercut ? Let there be : 

N = number of teeth in gear. 
n = number of teeth in pinion. 



nV^ N' + ^' _ 



N 
where we have undercut for/ less than 30. 

This formula is strictly correct for epicycloidal gears only. 
It is, however, used as a safe and efficient approximation for 
the involute system. 



20 



BROWN & SHARPE MFG. CO. 



DIAMETER INCREMENT. 



Rule. — The ratio being given or determined, to find the outside diameter 
divide figures given in table for large and small gear by pitch (P; and add 
quotient to pitch diameter. 







GEARS. 






GEARS. 




GEARS. 


RATTH 






RATJ'^ 






RATIO. 










Large 


Small 






Large 


Small 




Large 


SmalJ 


1 00 


1:1 


1.41 


1.41 


1.65 




1.05 


1.70 


4.40 




.45 


1.94 


1.05 




1.37 


1 42 


1.67 


5:3 


1.03 


1.72 


4.50 


9:2 


.44 


1.95 


1.07 




1.36 


1.43 


1.70' 




1.01 


1.73 


4.60 




.42 


1 95 


1.10 




1.35 


1.44 


1.75 


7:4 


.99 


1.74 


4.80 




.41 


1.96 


1.11 


10:9 


1.34 


1.46 


1.80 


9:5 


.97 


1.75 


5.00 


5:1 


.39 


1.96 


1.12 




1.33 


1.46 


1.85 




.95 


1.76 


5.20 




.38 


1.96 


1.18 


9:8 


1.33 


1.47 


1.90 




.93 


1 77 


5.40 




.37 


1.96 


1.14 


8:7 


1.32 


1.49 


1.95 




.91 


1.78 


5.60 




.36 


1.97 


1.15 




1.31 


1.50 


2.00 


2:1 


.89 


1.79 


5.80 




.34 


1.97 


1.16 




1.30 


1.51 


2.10 




.87 


1.80 


6.00 


6:1 


.33 


1.97 


1.17 


7:6 


1.30 


1.52 


2.20 




.84 


1.81 


6.20 




.32 


1.97 


1.18 




1.29 


1.53 


2.25 


9:4 


.82 


1.82 


6.40 




.31 


1.97 


1.19 




1.28 


1.53 


2.30 




.80 


1.83 


6.60 




.30 


1 97 


1.20 


6:5 


1.28 


1.54 


2.33 


7:3 


.78 


1.84 


6.80 




.29 


1.98 


1.23 




1.27 


1.55 


2.40 




.76 


1.85 


7.00 


7:1 


.28 


1.98 


1 25 


5:4 


1.25 


1.56 


2.50 


5:2 


.75 


1.86 


7.20 




.27 


1.98 


1.27 




1.25 


1.57 


2.60 




.73 


1.86 


7.40 




.27 


1 98 


1.29 


9:7 


1.24 


1.58 


2.67 


8:3 


.71 


1.87 


7.60 




.26 


1.98 


1.30 




1.22 


1.59 


2.70 




.69 


1.87 


7.80 




.26 


1.98 


1.33 


4:3 


1.20 


1.60 


2.80 




.67 


1.88 


8.00 


8:1 


.25 


1.98 


1.35 




1.18 


1.61 


2.90 




.65 


1.89 


8.20 




.24 


1.98 


1 37 




1.17 


1.61 


3.00 


3:1 


.63 


1.91 


8.40 




.24 


1.98 


1.40 


7:5 


1.16 


1.62 


3.20 




.60 


1.92 


8.60 




.23 


1.98 


1.43 


10:7 


1.15 


1.63 


3.33 




.58 


1.92 


8.80 




.23 


1.98 


1.45 




1.13 


1.65 


3.40 




.56 


1 92 


9.00 


9:1 


.22 


1.99 


1.50 


3:2 


1.11 


1.66 


3.50 


7:2 


.54 


1.93 


9.20 




.22 


1.99 


1.53 




1.10 


1.67 


3.60 




.52 


1.93 


9.40 




.21 


1.99 


1.55 




1.09 


1.67 


3.80 




.50 


1.94 


9.60 




.21 


2.00 


1.58 




1.08 


1.68 


4.00 


4:1 


.49 


1.94 


9.80 




.20 


2.00 


1.60 


8:5 


1.07 


1.68 


4.20 




.47 


1.94 


10.00 


10:1 


.20 


2.00 



Note. — To be used only for bevel gears with axes at right angle. 



PROVIDENCE, R. I. 21 



TABLES FOR ANGLES OF EDGE AND ANGLES 

OF FACE. 

The following four tables have been computed for the 
convenience in calculating datas for bevel gears with axes at 
right angle. They do not hold good for bevel gears with axes 
at any other angle. 

To use the tables the number of teeth in gear and pinion 
must be known. 

Having located the number of teeth in the gear on the 
horizontal line of figures at the top of the table, and the num- 
ber of teeth in the pinion on the vertical line of figures on the 
left-hand side, we follow the two columns to the square formed 
by their intersections. 

The two angles found in the same square are the respective 
angles for gear and pinion. The tables are so arranged that 
the angle belonging to the gear is always placed above the 
angle for the pinion. 



22 



BROWN & SHARPE MFG. CO. 



TABLE I 



Angle of Edge. 

GEAR. 







41 


40 


39 


38 


37 


36 


35 


34 


33 


32 


31 


30 


29 


28 


27 




12 


7341 

i6;i9 


73*18 

16*42' 


72*54 
17V 


72*26 
17*32 


72*2 

17*58 


7I-3V 

18*26; 


71*5' 
18*55 


70*34 
19*26 


70* r 

19*69' 


69*26 
20*34 


68*50' 
2»*io' 


68*12 

21*48 


67*3. 
22*29 


66*48 

23*12 


66*2- 
23)8 




13 


ltd 

17*35 


7.W 
18*1' 


71*34 

18*26 


71*7 
18*53' 


70W 

19*8.' 


70*9' 
I9>.' 


69V 

20*23' 


69*5 
20>s 


68'30' 

21*30 


67*53 
22*7' 


67*15' 

22*45 


66*34 
23*86 


65*si- 
24*9; 


65*6' 

24;S4;- 


64*17' 
2543; 




14 


71 V 


70*43 
J9*I7' 


70*15 

I3%s 


69*46 

eo*M.' 


69*.6 

2044 


68*45 
21*5 


68'i2' 

2(*4« 


67*i? 

22*23 


67'o' 
23*0' 


66*23 
23*87 


65*42 
24*.6; 


64*» 


64*H»' 
25*4« 


63*26' 
26*34 


62y 

27*84 




l5 


69V 

eoV 


68*2^ 

20*34 


68'se 


68*28 
21*32 


67's6 
22*4 


67*n 

22*37 


66*48 

2312 


66-12 

23*48 


65*33' 
24*27 


64y 
25*7- 


64',o' 
25*50 


63*26' 
26*34- 


62^9 
27*21 


61 '4» 


60*5^ 
29*» 




16 


68V 

21*3 


68'» 

81*48 


67V 

ezV 


^7*.o 
22*50 


66*37 

23*23' 


6e'2' 

23*58 


65*26' 

24*34: 


64*48 
25*12 


64*8' 

25*58 


63*26 
26*34 


62*42 

27*»' 


61*56' 
28*4 


61*7 
28*53 


60*15 

29*45 


59'er 
30V 




17 


67« 

2e*3i' 


66*« 
23*2 


66*27 
23*33 


65*54 
24*6' 


65*13' 
844.' 


64*43' 

25*17' 


64*6 

25*54 


63*26 

26*34 


62*45 

27*is 


62* r 

27*59 


61*15 
28*45 


60*28 
29*32 


59*37 
30*n 


58*44 
31*6 


57*48' 

32*12 




18 


234e 


6S'4« 
24*M 


esv 
24*4«; 


€4's3 
25*2. 


64V 

25*56 


63'W 

26*34 


62*47 
27*13 


62*6 

27*54 


6l'25' 
28*57' 


60*36 
29*22 


59"5r 
30*9 


59*2' 
30« 


58*10 
3V*M 


57'',6 
32*4^ 


56*.9' 
334.' 




19 


65*e 
2**« 


64*M. 
25*« 


64*2 

25*58 


63» 

26*84' 


62*49 

27*11 


62*10 
27*56 


61*30 
28*3<J 


60*48 

29*12 


60*4 
29*5^ 


59*.e 

30*42 


58*30 

81*30 


57V 
32*er 


56*46 
S3*<4 


55*51- 
34*9' 


S4'5« 
35*8' 




20 


64-0- 
26*0 


63*26 
26*3*' 


62*5.- 
27*9' 


62*14 
27*46' 


61*37 
28*23' 


60*W 
29'» 


60*15 
29*45 


59*32' 
30*28 


58*47- 
31*13- 


58*0' 
32*0' 


57'k)' 
32*50 


56*19- 
33*41' 


55*» 
34*3^ 


54*28 
35*32' 


53*28 

36*32* 




21 


620 
27*7" 


e2*« 

274i 


61*42 

28V 


61*4' 
28*56 


60*25 

29*3s' 


53*45' 
SO*.s' 


59"a- 
30*58' 


58*8 

31*42 


57V 

32*26 


56*43 

33*17 


55*53' 
34*7' 


55*'o- 
35*0' 


54*5' 
35*5S 


53*7 
36*53 


52*8 
37a- 




22 


61*47 

28*13 


28*43 


60*34 
29*% 


59^* 
30*4 


59*.5' 
30*4S 


58*34 
31*26 


57*5.' 
32*3' 


57V 

32*si 


56*19 
33*41- 


55*29 

34*31' 


54*38- 

35*22' 


53*45 
36*15' 


52*49' 

37*11' 


51*50- 
38*10 


50'43' 

39*1 r 




23 


60V 

e9'« 


60*6 
29'si 


59*28 
30*32 


58*43 
31*11' 


58*8' 
31*82 


57m' 
32*3s 


56*41' 
33*13' 


55)5 
34*s' 


55%' 

34*53 


54*«' 
35*42 


53*26 
36*34 


52*3.' 
37*29 


51*35 
38*2S 


50*36 
39*24 


4^34 
40*86 




24 


53*39 
30*er 


59*2- 

ao's* 


58*es 
31*37 


57*44 

32}6' 


57''e' 
32*se' 


56*9' 
33*41' 


55*33 
34*27 


5447 

35*13 


53*58 
36*2 


53*7 
36*53 


52*15 
3745 


51*20 

38*40- 


50>3 

3957 


43*24 
40*36 


48*2^ 

41*36 




25 


58V 
3lV 


58*0' 
32*0' 


5720 
32*40 


56*40 
33*eo' 


55*57 
34*3 


ssV 
34*47' 


54*28 
35*32' 


53*40' 
36*20 


52*51' 
37*9' 


52*g- 
38*0' 


Si"? 
38*S' 


50*12 
39*48- 


49*K. 
40*46 


48*.4 
41*46 


47*12 
44*48 


'4. 

O 


26 


5737' 
32*23' 


56*S8 
33*e' 


56*« 
33*41 


55*37 

34*23' 


54*S4 
35*6 


54*.d 
35*sd 


53*24 

36*M 


5296 
37Vv 


5.*4^ 

38*14 


5054 
39*6 


50*,- 
39*59 


49*5 
40*55 


48*7 
41*53 


47*7' 

4e*ra 


46*5 
43*55 


1<^ 

SI 


27 


56*38 
33*« 


ssW 
34* r 


S5*i8- 

34*42 


S4V 
35*M 


53*53 
36*7 


53*7- 
36*53 


52^21 
37*39' 


51^3 
38*27 


50V 
33*17 


49"5. 
40*9' 


48"57 
41*3 


48''o' 
42*0 


47*3' 
42*57 


46'2 
43*56 


4-5* 




28 


SS4*; 
34*20 


55*0 
35*0' 


S4*« 
35*4. 


53*37 
36*23 


52*53 
37*7' 


52*8 
37*52 


5'> 
38*40 


50*32 
39*28 


49«4r 

40*13 


48*46' 
41*12 


47''55 
42*5 


46>6 
43 2 


46*0 
44*0 


45* 






29 


54W 
35*6 


35*S7 


53*22 

36*38- 


S2'^3S 
37*er 


51*55 
38-S 


51*9' 

sa'si' 


50*21 
39*39 


49*32' 
40*28 


48-4.' 

41*19 


47*49 
42*11 


46V 
43*6- 


45^ 
44*2' 


45* 








30 


S3*4i 

36*«' 


53*7- 

36*53 


52*26 
37*34 


5.*42' 

38*8 


50*56 
38*t 


50*ie 

39*46 


49*24 
40*36 


48*3S 
41*25 


47*43 
42*17 


46*5. 
43*9 


45-56 
44*4- 


45* 








31 


37V 


58*.3 
3747 


5l*3»' 
33'w 


50*48- 
39*2 


50*2 
39'w 


48*.6' 
4044 


48*26 
41*32 


47*33 
42*21 


46*47 
43*«3 


45*54 
44*6 


45* 








32 


52*2' 
37*« 


51*20 

38*40 


50*38 
39*22 


49*54 
40*6 


49*9 
40*51 


48*22 
4i'3»' 


47*34 
42*2t' 


46*44 

43*16 


45*53 
44*7' 


45* 








33 


51*10 
38*50 


50*2^ 
39*31 


49''46 

40*4 


40*58 


48*16 
41*44 


47*2» 
42*21 


46''4.' 
43-.0 


45*5. 
44*9 


45' 








34 


50*20 
39*40 


49*3« 
40*« 


48^Si 

41*5' 


48'..- 
4t*4a' 


47*25 
42*35 


46''38 

43*22 


45^0 
44*10' 


45* 








35 


43*3. 

40*29 


48*^ 

41*18 


46*s 
41*55 


47*« 

42*39 


46*35 
43*25 


45*48 
44*tt 


45' 




• 




36 


48« 

41*17 


48^0 
42*0 


47''a 

42*43 


46*33 

43*27 


4547 

44* iW 


45* 








37 


47'k» 
42*4' 


47'',4 
42*4* 


46*30 
43*30 


45!U 
44*»4 


4S* 








3a 


4^10 
42*S9 


46'2» 

43*3e 


4«^4S 
44*15 


45' 








39 


46*2. 
43*31 


45*43 

44*17 


45- 








40 


4542 
44*,B 


45' 








41 


45- 















PROVIDENCE, R. I. 



23 



TABLE \— (Continued) 



Angle of Edge. 



GEAR. 





26 


25 


24 


23 


22 


21 


2U 


19 


18 


17 


16 


15 


14 


13 


12 




12 


65-.. 


64'2Z 

25°38 


63*ZG 
26a4 


62*27 

27°33 


6i'e3 

28*37 


60*15 

29*45 


59-2 

30*58 


57*44 

32*16 


56 '19 
33*4i' 


54*47 

35*13 


53*7' 

36*53 


51*20 

38*40 


49*24 
40*36 


47"i7' 

42*43 


45* 




13 


63*26 
26V 


62*31 

27*29 


61 '33 
2827 


eoSr 

2^*29 


59'2S 

30*35 


5e*»4 

31*46 


56-58 
33*2 


55'37 
34*23 


54*10 
35*50 


52*36 

37*24 


50*54 
^9*6 


49' 5 

4055 


47*7 
42*53' 


45* 






14 


6l'4E 
28*iB 


604S 
29*15 


594S 
30*15 


5840 
31 '20 


57*32 
32*28 


56*19 
33*41' 


55*0 
35"o 


53*37 

36*2i 


52*6 
37m 


50*32 
39*28 


48*48' 

4ri2' 


46 58 
43'e' 


45- 






15 


60*.- 
29*53 


59- 2 
30*5* 


S8"o 
32*0' 


56V 
33*7 


5543 

34'|7' 


54-28 

35*^2 


53*7' 
36*53' 


5lV 
38*18 


50"i2 

38*48' 


48°i5' 
41*25 


46*51 
43"9' 


45' 






16 


58*21 
31 V 


57'e3 

32*57' 


5fc*l9 

33** 1 


55*11' 

3**49 


53-sB 
36*2' 


52*42' 

37*18' 


51 M 

384<i 


49-S4 

40*b 


48*22' 

4lV 


46*44 

43*16 


45* 






17 


56V 

33*ir 


55*47 

34*13 


54'4r 

35•l^• 


53*32 

36*28 


52- .8 

37*42' 


sro- 

33*0- 


49» 

40» 


48*n' 

4143 


46*38 
43*22 


45* 






18 


SS'it 
34%e 


54-.5 
35*4s 


53-7 
36*5i 


5.V 
38*3' 


5043 
39*17 


49W 
40*36 


48'« 
42*0 


46*33 
43*27 


4-S* 






19 


53-5, 


5246 

37*rt' 


51*38 

38k 


50*26' 


43*11' 
4043' 


47*52' 
4?*8 


46*28 

43*« 


^v 






20 


52'es 

37°*!: 


38%(; 


50*R 
39*48 


48*53 

4l*i' 


47*43 

42*17' 


46*24 
43*3* 


45- 






21 


38'56' 


49*58 
40*2' 


48*48' 
41 V 


47*36 

42'm; 


46'£d 
43°40 


45- 






11 


4946 

40*14 


48*« 
4I°2(' 


47-2, 

42*31' 


46*.e 
43V1; 


4-5° 






23 


48*3» 


47W 

42*37' 


4€a 
43%7 


^s' 






24 


42*43 


46*10 
43*50' 


45' 






25 


46*7- 
43's3 


4.5* 






26 


4.5' 







tan a„ 



N. 



tan.,= N, 



{See page 13.) 



24 



BROWN & SHARPE MFG. CO. 



TABLE 2. 



Angle of Edge. 

GEAR. 





1 — 


72 


71 


70 


69 


68 


67 


B6 


65 


64 


63 


62 


61 


60 


59 


SB 


57 




12 


9'tT 


80'M 
9V 


eo« 

9*44 


86*8 
9*52 


79*S6 

loV 


79*6t 
10*9 


79*42 
10*18 


79*32 
io°ae 


73;ri 

10*37 


79*.3 
10*47 


79*3 

10*57 


78*52 

ii°e 


78>,- 

11*19 


78*36 
11*36 


78*19 

1 1*4.' 


78*7 
11*53 




13 




79V 
10*23 


79*2» 

I0*3l" 


79*20 
10*40 


79*..' 

10*49 


79;. 
10*59 


18*5. 
11*9' 


78*4.' 
11*19 


78*3.' 
11*29 


78*20 

11 V 


7^9 
11*51 


77*58 
12*2 


77V 
12*14 


77V 

12*26 


77*22 

12*38 


77*9 
12*51 




14 


79-<, 


78-5. 

11*9 


784.' 
II* « 


76''32 
11*26 


78*22 
M*3i 


76*.. 

11*49 


78* .' 
11*59 


77*51- 
12*9 


77*40 

12*26 


77*28 
I2*3i 


77*,7 
12*43 


77*5 
12*55 


76*5i 
13*8 


76-» 

13*2.' 


76*26 

13'3i 


76*12 
13*48 




15 


78'l4 


78%' 

11*56 


77*9V 

12*6' 


77V 

12*16 


77*34 
IE*i6 


77*23 
12*37 


77'.2 

12*48 


77*o- 

r3*o' 


76*48 
13*K>' 


76*36 
13*»* 


76V 

13*3^ 


76*11' 
13*49 


75*59 
\4°e' 


75*4^ 

14*16 


7^36 
14 36 


7i*,5 
14*45 




16 


77-26 


77-S 
12V 


12 53 


76*57 
13*3' 


76*45 
13*15 


76*3» 
13*26 


76*22 
13*3^ 


76*10 
13*50 


75*58 
14*2 


75V 
14*15 


7S*3i 
14*28 


75*« 
14*42 


75%' 
14*56 


74V 
15*11 


74V 
15*25 


74-,9 
15*41 




17 


76*43 
13V 


7^K 
I3W 


76*2.' 
I3*M 


76*10 
13*50 


75*58 
14*2 


75V 

I4*ti 


75*33 
14*27 


75*2. 
14°m' 


75°e' 

14*62 


74*54 
15*6 


74^40 
15*20 


74*25 
15*35 


74*1.' 

15*49 


73*56 

16*4 


73*40 
16*20 


16 36 




18 


75^58 
14° t 


754S 

14° KV 


75*3S 

I4*2S 


75*23 
I4°37 


75°.6 

14*50 


74*59 
15*2 


74*45 
15*15 


74*3, 

15*29 


74*17 

15*43 


74*3 
I5*5i 


73*49 
16*..' 


73*33 
16*27 


73*.8 

I6*4i 


73*2 
16*58 


7245 
17*15' 


72*29 
17*31 




19 


7S^« 
»4%7 


7S'.- 
»4'59 


74'-« 
15*1 1 


74^3* 
15*24 


74*23 

15*37 


74*Ki 
15*56 


73V 
16*4 


73V 
16° 18 


73*28 
16*32 


73V 
16*47 


72*5S 
17*2' 


72*42 
17*18 


72*20 
17*34 


72*9 
I7°5l' 


71*52 
18*8 


7l'3» 
18*26 




20 


74-« 


74'.6 
I5*W 


IS 57 


73*50 
16*10 


73*37 
16*23 


73*23 

16*4;; 


73*9 
16*51 


72V 

17*6 


72*39 
17*21 


72V 
17*37 


72*7 
17'53 


71*5.' 
18*9' 


71V 
18*26 


71*16 
18*44 


70''59 

19* r 


7(f46 
19*20 




21 


73-45 


73*32 


73*.8 
16*42 


73*4 
16*56 


72*50 
17*10 


72*36 
17*14 


72*2. 

17*39 


72V' 

17*5f 


71*^50 

l8*.o 


71-34 
18*26 


71*17 

18*43 


71*0' 
l9*o' 


7043 
19*17 


70*24 
19*36 


70*6- 
19*54 


69V 
20*24 




22 


73- ,• 
16*59 


72t,7 
I7''l3' 


72-« 
17*27 


72*,,' 

17*4.' 


72*/ 
17*56 


71*49 
18*11 


71*34 
18*26 


71*18 

!e*42 


71*2 
I8*5R 


70*45 
.9*15 


70V 
19*3i 


70°.6 
19*50 


69*52 
20*8 


69*33 
20*27 


63''l3 
20*47 


68*» 
21*6 




23 


72*^' 

»7%3 


72°3 
17 $7 


71*40 

18*11' 


71*34 
18*26 


7I'«' 
18*41' 


7J*3 
18*5^ 


70*47 

i9*a 


70*30 
I9*3«i 


70*.i 

19*46 


69*57' 
20*3 


69>9 
20*21 


69-20 
20*4<i 


69V 
20V 


68*42 
21*18 


68*22 
21*38 


68*2 
21*58 




24 


71-3. 
16* tt 


71*« 
I84t' 


7l*s' 

18*55' 


7<f43' 
19*..' 


70*34 

19*26 


70*17 

19*43 


70*.' 

19*59 


69*4i 
20*16 


69*26 
20*34 


69*9 
20*51 


68V 
21*10 


21*29 


68*.i 

21*48 


67*52 
22*8 


67*3.' 

22*29 


67-,o 
22*50 




25 


70*si' 
J9*9' 


70-3. 

•9*24 


70*ti 
19*39 


70*5' 
19*55 


69;49' 

20°ii' 


69*32 

20*2» 


69"\5 
20*45 


68*57- 
21*3' 


68*40 

ziV 


68*2.- 
21*39 


68*3- 
21*57 


67*43 
22*17 


67*23 
22*37 


67*2' 
22*58 


66*4.' 
23*19 


66*19 
23*41 


5?; 
o 


26 


70*9 

t3*5l 


69*53 
20*7 


69*37 
20*« 


69*2.- 
20*39 


2056 


68*46 
21*12 


68-30 
21*30 


68*12 

21*4* 


67V 
22*6 


67V 

22*26 


67;.5 
22*45 


66*55 
23*5 


66*34 

23*26 


66*13 

23*47 


65*si 
24*9 


65V 

24*31' 




27 


20*SJ 


6SV 
20*50 


68>4 
2I*6 


68*38 
21*22' 


66*2^ 

21*40 


68*3 

21*57 


67*45 
22*15 


€7V 
22*3i 


67*8 
22*52 


66V 
23*12 


66*28 
23*32 


66-7' 
23*si 


65*46 

24*14 


65V 
24*35 


65*2 
24*58 


64*39 
25*«.' 




28 


68%i 
21* 15 


b8'» 
21*31 


6S-12 
2(*4e 


67*55 
22*5 


67*37- 

22*a 


67'.9 
22*41 


67*,' 
22» 


66*42 

23*18 


66*22 

23*38 


66* 2; 
23*5i 


6S*4i 

24*18 


6tf2i 
24*39 


€4*59 

25*.' 


64*37 

25*23 


64* w 
25*46 


63-S6 
26*10 




29 


68V 
2l'56 


67^47 

22*13 


22*^ 


67*12 
22*48 


66V 
23*6 


66*36 
23*24 


66*.7 
23*43 


6^57 
24*3 


Og'37 

a4-a 


65*,6 
24*4^ 


64V 
25*5 


64V 
25*26 


64V 
25*4i 


63V 
26*.o' 


6^26 
26*3* 


63*2 

26*58 




30 


22*37 


6r6 
22*54 


23*12 


66*30 
23*90 


6^,i 
2348 


6^52 

24*8 


65*» 
24*27 


6S*h; 
24*46 


25*7 


64*32 
25*28 


64*.o' 
25*50 


63*4,- 
26*11' 


63*26 
26*34 


63-3* 
26*57' 


62*3,' 
27*21 


62*14 
27*4« 




31 


66V 
23*»8 


66'2S 
23*3i 


66V 
23*si 


65*46 
24*12 


65*2^ 
24*3. 


65*10 
24*s« 


64*S« 
25*10 


64*30 
2S''30 


64*9 
25*5. 


63V 

26*«; 


63'2« 
26*3i 


63V 

26*57 


62*40 
27*20 


62*.ri 
Zf\i 


6IV 
28*7 


6i:28 
28*3* 




32 


66V 
23'h« 


65*44 

24*.e 


6S-20 

24*34 


6S-7' 
24*53 


64*4^ 
25V 


64*28 
25*32 


W*8 
25*S2 


61^V 
26*13 


63-2. 
26*>i 


63-^ 
26*56 


62V 
27*« 


62*,9 
27*4,' 


61*56 
28V 


6rn 

28>8 


61*7' 
28-53 


6(^4. 
29*« 




33 


6523 
24*37 


65V 
24*56 


64*45 

25*.S 


64*2i 
2S*S^ 


64*7' 
25*53 


63*47 
26*13 


63*26 
26*34 


26*55 


62*43 
27*17 


27*« 


61*58 
28*4 


6I'3^ 

28*ti 


6l*ii' 


60>7 
29*3 


60-2,' 
29*39 


59*56 
30V 




34 


644J 
Z5*.7 


2535 


64*.' 
25*55 


26*14 


6326 
26*34 


26*55 


62*45 

27*is' 


62V 

27*37 


62*.' 
27*5? 


61*38 


61*15 
28*45 


60V 
29*8 


60*28 
29*32 


60*3 
29*57 


59V 
30*23 


59*.r 

30*49 




35 


es'ss 


w;4^ 

26*15 


63*2fc 
26°3* 


63; 6 

26*54 


62*46 
27*14 


62*25 

27*1$ 


62*^' 
27*56 


61*42 

28*18 


6.*„' 
28*41 


tO*S7 
29*3 


60V 

39*27 


60*9 
29*51 


59*45 
30*15 


59*19 
30*41 


58V 
3lV 


58*27 
31*39 




36 


63*26 
26'>i 


2653 


62V 
27*13 


62*27 
27*33 


6<< 

27 54 


61*45 

28*45 


6)V 
28*37 


61*,' 

28*59 


60>8 
29 2i 


60*.5 
29°4i 


59*51' 
30*9' 


59*27 
30*33 


59*2 

30*58 


58V 
31*23 


58*10 
31*50 


57*43 
32*17 




37 


27-.2 


62-2S 

27» 


62^8 
27*52 


6l*4i 
28*12' 


6»*27 
28*33 


6»*b' 
28*55 


6OV 
29*16 


60*t,' 
29*3i 


S9*si 
30*2 


59V 
30°2i 


59*10 
30*50 


58*4; 
3I*|A 


58V 
31*46 


57*54 
32*6 


67*2« 
32*31 


57* .• 
32*5» 




38 


2749 


&l*5.' 
28*,' 


fel*3<i 

2e*s<^ 


6l°9' 
28*5.' 


eo*4a 

29*1» 


60*2i 
29 S4 


60*^ 
29*56 


30*.9 


30*4i 


S8V 
31*6 


58*^6 

31*30 


58*S' 

31*55 


57V 
32*2l' 


57*.3' 
32*47 


56'4« 
33*14 


5^» 

,3V 




39 


61*33 
2B*27 


61*13 
28*47' 


60*53 

29*7 


60*31 
29*29 


60*10 
29*50 


59*48 

S0*« 


59-25 
30*35 


59V' 

30*58 


58V 

31*21 


58*^ 

31*4^ 


57*56 
32°>6 


57*2^ 

32*36 


56*58 
33*2 


56*3i 


5^6 
33'9» 


55*,7 
34*23 




40 


60's7 

z4i 


60^56 
29*4.- 


60^5 
2945' 


30*7 


59*32 
30*28 


59*10 
30*50 


5tf47 
31*13 


58*« 

31*3i 


58*0- 
32*0' 


57*35 

32*2fi 


S7*.6 
32*50 


56*44 
33*16 


56V 

33*4..' 


55*52 
34*8 


55*24 
34*35 


54*57- 
35*3 




41 


60*26 
29*40 


60-0- 
30*0' 


59*3, 
30*21 


59:.7; 

30 43 


58*55 

31*5 


58*3i 

31*28 


58*9 
31*51 


57V 
32*5 


57*z.- 
32*39 


56*57 

33*3 


56V 
33*?* 


56;6' 
33*54 


55*39 

34*2 1 


55*.i 
34*48 


54*V 
35*16 


54*6 
35*44 




4^ 


S9-4i 
30.S 


59*» 
30*3^ 


59V 
30*57 


58V 
3l*ao 


58^,8 


57*SS 
32*5 


57V 
32*28 


57*8 
32*si 


56*43 
33*17 


56*19 
33*4i' 


55*53 
34*7 


55*27 
34*33 


55*0' 
35*0' 


54*33 
35*27 


54*5' 
35*55 


53*37 



PROVIDENCE, R. 1. 



25 



TABLE 2.— {Continued.) 
Angle of Edge. 

GEAR. 



5655545352 



SI 



50 



494847 



4645 



44 



[43 42 

as 7? 



1? 
13 

14 
15 
16 
17 

18 

19 
20 

21 
22 
23 
24 
25 
26 
27 
28 



77 m 
12° 6 



77« 

12*32 



I2%5 



7^4€ 
13" 14 



76*30 76V 
I3*s6 134* 



75;se 



7S"4l 

14* f9 



l^ii l^A 



74"4« 



75'tt 7S»e 
14*34 14*52 



35 15*57 



IC^s 17*12 



7642 

I3'» 



76'l3 
13*47 



75' 
14*^2 



7^44 
14*18 



74-51 
»5*9' 



74« 
15*28 



74*13 
15*47 



7363 



7S'S6 
14*2 



7?43 
»4'l7 



75" 
14*32 



75^ 
14*48 



74*56 
15*^' 






74*21 7^ 
IS'm »*»7' 



73*44| 73*25 



73^ 

16*56 



72 
I7V 



72*21 



7»^7I''34 

fix te« 

>*46 70\\ 



75V 
l5*o 



M44 

15*t6 



74» 
IS*3i 



74 12 
tB'49 



73S5 
W*5' 



73] 
16*231 



3I7-73 



^18 78*59 
16*42 17 



72 39 
«7"2I 



72j8 
174Z 



7I°66| 
18 



71^34 
I8*8(i 



71 to 

18*50 



l» »4 



^26 

19' 



74;3 
15*57 



16 13 



73' 
1^ 



72V4 72' 



35 
17*25 



72" si 71*55 

.e"*5' 



71*13 70*52 
e*47 19*8 



7134 
18*26 



7 1" 12 
1848 



70^49 
19*11 



70*30 70V 
19*30 19*53 



70" 

19*591 



C93S 

20* 



20*5) 



737 
16*53 



7249 
17'„ 



72*31 
17 29 



72-13 
17*47 



7154 

18" 6 



7l''34 
19*26 



6943 
20"l7 



e9"l7 



68*52 



6686 



^7M 



1749 



71*53 
18*7 



18*26 



71-15 
18*45 



7o*»i; 

19*6 



70-33 

19*27 



70*2 69*50 
19*48 20*10 



69 3 
20*57 



£8 
2l'^2« 



67 
22*^15 



66*48 



22 43 

67-3<U67-6 66-38] 66° 10 
22*26 22*M 23*22 23*50 



71*15 
I8*4S 



70 57 

i3"r 



70-371 

19 



7017 

19*43 



6956 

20* 



69 



69*12 6e 
20*48 21 



68 25 
21 



67 59 
22*1 



65*39 



70r2i 
I9*» 



69'4i 
20*19 



69J9 
20*41' 



ee's? 

21*3 



68 35 66 12 

2I*48|22*I2 



6748 67'23l6^S7 

23% 



i?2 
23* 



65*33 65*3 
5«| 24*27 24* J7 



6»*t6 

2oVi 



69°o 
20''s4 



«8'4!; 

21*1 



68*0 
22*0' 



6737 

22*23 



67*13 
22*47 



23*12 



65^55 
24*5 



65*28 
24*32 



64^59 
25*1 



64-29 63 58i 

2S*ai 



6326 
2<r34 



67^27 

22u 



22*561 



23*201 



6833 
21*27 



67 
2St\i 



66rs 

23*45 



65*49 

24*11' 



24*37 



64 55 
25*5 



64"^ 
25*34 



63*57' 
26*3 



62*5M€22I 
27*» 



27 6 
75- 



67*41' 
22*19 



6rt8 
22*42 



66-5S 
23*5 



23 



66*8 

23*52 



65*44 

24*16 



63'w 

2442 



64*51 
25*9 



64-a4j 
25' 



Z6*S 



63-26 
26°3* 



62*56 
27*A 



62*a4 
27 



6i*sc 6r 



36 28 » 



2S4e 
29-4i 



6648 

23*12 



66 26 
23*34 



66' I 
43*58 



65m 

24*22 



6S*|4 
24*46 



64 

25*12 



48 6422 
25*3 



63*54 

26*6 



63*26 

26*34 



62^57 
27*3 



62*27 

27*33' 



6r5« 
28*4^ 



6r23 
28*37 



6Z*S< 
27*2 



29 to 

59jso 
Zd'stiSo'id 



-^z 



59*14 
30*46 



65*57 

24*3 



6533 

2**27' 



65*9 
24*Si' 



64-45 
25*15 



64 20 
25*40' 



6353) 

26*7 



63 a6 
26*54 



62*w 
27*31 



61-59 
2S*i 



61-29 
28*31 



60*57 

29*3 



65'6 

24*9^125 



6442 

'e 



6<I8 
2^- 



63*52 
26*8 



63*26 
26*34 



6^99 
27* » 



623t 
27*291 



62*3 
27*57 



6133 
28*27 



613 
28*57 



6d^ 
29*29 



5959 
30*1 



59*25 
30*95 



58 50 



S?C 



64' 
25*44 



63*2i 



27 
27*51* 



62 34 
27*26 



62!fi 
27*s4 



6138 61*8 
'22 2«*S2 



60^38 
29*23 



29*53 



59-35 

30*25 



59°» 
30*58 



58*28 
31*32 



32*7 



33*4 



5?5 



34 2659 



6236 
27*24 



6l4* 

26*18 



61 14 
26*46 



60*45 

29*15 



6^ 
29*46 30' 



59 >3 
30*47 



58^ 
31*20 



31 53 



32*28 



34*87 



29 



62*37' 
27*23 



6^12 
27*48 



61* 
28*15 



6119 
28*41 



60~S1 
29*9 



60 2J 

29*37 



59*53 59 



30 7 



30 37 



58 52 

31*8 



58 

31*41 



19 57*46 57 12 



3248 33 



56*37 
n 



34* o 



30 



61*49' 
u' 



6r» 

28*37 



60 
29*3' 



60*29 

2931 



29*99 



59m 
30*28 



59*2 



S^ 



3058 3128 



580 
32*0' 



57*27 

3233 



56^ 
33*7 



55^.9 
33*41 



55-45 



sy 



34*0 35*32 



31 
32 



61' I 
28*18 



6036 

29*** 



60 6 
29*S4 



59 ta 
30*4^ 



S8*42 
31*.8 



58*12 574i' 
31*48 32*19 



S7'8 
32*52 



57*2i 56*52 
32*37 33*8 



56*3^ 
33*24 

34*15 



5^ 
33*69 



SS*26 



35*10 



35*48 



53*34 

36*26 



59-48 

90*12 



B9aj 
30*39 



56 a 
31*8 



58 
31*26 



56*3* 56*2 

33*26 33*58 



56-19 
33*41 



34*49 



54"3s 
35' 



53*5»| 
36*2 



52^ 
36'm 37*1© 



33 



59 29 

30*31 



59*2 
30*58 



34 58' 
26 31* 



57'36 
32*a» 



57-6 
32*54 



30 54-56 



54*2 
35*39(36 15 



53 8 
36*52 



37*3. 



Sl'so 
38%' 



34 



58*44 Se 16 



57 46 57 19 



31-16 



31 44 32 12 



3241 



56 49 
33*11' 



5619 
33*41 



55*5 
34*13 34*45 



5S*o' 54*» 
35*0 3B*3i 



54-41 
35*19 



54 7 
3S*S3 



53-32 

36*28 



5252 

a 



52*18 

37*42 



51*40 

38° 20 

50^ 



390' 

SoTit' 



35 



58 
32*0 



57*3i 
32*28 



32 57 



56'33 

33*27 



56 3 
33*57' 



55J3? 
34*28 



53*54] 

36*8 



53*26 

36*40 



52*44 

37°i6 



52*8 

37''b2 



STso 
38*30 



39 



50*4 
39'^ 



36 



S64S 



56*19 
33*41' 



55*49 

34*11 



55*18 



5447 

35*13 



33*e8|33'S6|34' 



54°iB 53*42 
35*45 36*18 
53*30 Sl'tt 



53*6 
36*52 



5233 
37*27 



5167 
38*3 



61-20 
38*40 



5cr43 

39*17 



5(7S4«*5«|49' 
3646139*25 40%' 



S6 48 36 

'n 4eV 
40*49 44*23 



37 



55» 

» 



55s 

34*56 



54-34 

35*26 



54-2 



52*»3 

37*37 



51'47 

38*13 



48 li 47 §8 
'zi 42 

4^47* 



38 



55si 
34*9 



34*39 



15^52 
35*8 



5423 

3^37' 



53*51' 
36*9 



3»42 



52W^ 
37 



3748 



51*38 
3e°2« 



38*57 



50^27 
39*33 



4949 

40' 



49 II 

Ao'49 



39 



SV'm 



3451 



5?ib 
35*5fi 3621 



53 
36*53 



52^3*1 
37 



•3' 51": 
57 38°: 



50 54| 
39*6 



SO 19 
394 



494« 
40*18 



S 
40"65 



46 t7 
4I*3J 



I2J42S3 



40 



54 28 srse 

35*32 36 2 



53 28 5258 
36%2 37*2 



5226 
37*34 



5I°M 

38*6 



20 SOV 

38*40 39*14 



47^ 47»& 4? 



3948 40 24-41 I 



»3»42 k; 



L°ss' 43*38 
m43;I 



41 



3612 



17 
3643 



5248 

37*12 



5216 
37*44 



51 45 
15 



50 
384« 39 



39 50 
2139 



49 
40'3b 



30 46 54 48 r? 



47 40 
42*20 



42 59 



3$ 4419 

44*20 ^^ 



42 



53*6' 
36* 



51*4 50*3«49|s8 491 
S2|37'22i37'S2i38>4 38*86 39*28 40*1 40* 



92*38 52-e' 



5r3« 



4649 
H 



36 41 II 4< 4.7 42 84 43 I 



46jzo 
43*40 



26 



BROWN & SHARPE MYG. CO, 



TABLE 3. 



Angle of Face. 

GEAR. 



41 



40393837363534 



333231 



30292827 



12 



13 S7 
70' 



13 57 

TOail 



70*6' 



14 39' 

69*3 



15 
^69* 5' 



\St4 

fe8*3« 



15^9- 

67'<59 



16 1^ 
672 



16 43 

66*-i^ 



7l8 

fefc*5 



I7a3 
65i3 



18 .9 



18 5 
63'4d 



27 20 * 



63 



3' 62' 9 

49*7 JSVo 



13 



Wss 
69V« 



5'. 17 



I5'4B 

66' 



I6°a 

67°43 



6°si 
67*9 



17 
6633 



175^ 

6556 



18* 
64*S4| 



I9°57 

63*5 



20j 
62'i4 



14 



16 34 
68° 



16 59 

6 729 



I7\4 

66^6 



1750 
66°«£ 



65-47 6<S'9 



64-°30 6348 



20to 



206 
62°20 



21 34] 

6;''a2 



25*to 



SS* 

i5^ 



5247 

3I''40 



15 



17^ 
6645 



13 le 



18 
6540 



19"! 



1 9=40 
64-''e* 



20 "11 
63°4« 



204.( 



2/ 
62»4 



2 1 "53 

<-ra9 



22°3. 
60«( 



23 10 
60°a 



2S'S( 
59*a 



24^ 
58'3 

5642 



16 



I8°4e 
66 "4. 



19 

6533 



I9°aj 

64°«« 64°2>|6S4« 



21 -3 
63j7 



23sl 
60*14 



25^0 

58 6 2 



24*4 2 
5t\ 



17 



I9V4 
64' 



2024 
64'io 



62 



22 24 

61 ffo 



6I°9 



24 

594-0 



2 759 
5'6*|9 



27^7 



18 



2i¥ 

63*3 



22^ 
62341 



22%8 



23 • 

61*17 



2343 

60V 



24 „ 

59"5« 



25 
58 20 



26)5 

^7 31 



26 57 
S6°39 



Z7%x 



28*9 
5449 



£350 
3043 



28 

32'<» 



19 



22 

|6g.'3«| 



23 20 

61*24 



23 52 

6044] 



24i^ 
60*" 



Ii5 

58 54 



2537 
5837 



26°5» 
57% 



2 7 38 

56' 



28 22 

5521 



as"* 

54'2« 



2 9 56 
53 



5(24 

33'« 
50*4 



20 



23' 

6130 



24 

60*53 



2432 

60*14 



2^ 

59*34J 



26°)6 
58* 



26 55 
5725 



27 34 

5 6 S3 



28(5 

5549 



28°58 

54*6 



29°44 

54"* 



30 31 
53*9 



51 9 



21 



24 

6025 



25 (o 

59°46 



59S 



26 

58-26 



26"53 
574 



27 30 
5-/o 



28(0 
56*/4 



29'8t 

54°36 



30(7 

53*8 



52*50 



3(52 
51*3 2 



3244 

50*53 



33*3* 34* 
49*50 484 7 



22 



2546 

5920 



26-9 
56' 



2653 
58° 



27^7 

57*8 



28*5 
56*3< 



28°43 

55*5 



29'i>i. 
554. 



30 5 
54*7 



304B 
53*26 



3l°.34 
Sfc*32 



32 4 2 

5/*a 



49*4 



35*54 

47l2 



35*20 3?IT3ir^ 



46^0 

3a'28 



23 



26°«2 

58 (6 



27°n 

573B 



28* 

56*56 



29(4 
5530 



2953 



35»5 

53*57 



3(*.B 
538 



32 

52-,s 



324« 
5/°a4 



3336 
50*28 



34-27 
49*29 



37% 



46* a 45 II 



2A 



27s7 
57t5 



28 3 

56*35| 



29 

5553 



294; 
55*( 



54*2« 



3I°2 



3I°45 

52*5 r 



32*28 

52*2 



33 V 
SI 



34°. 
50*5 



3450 
49*20 



35-41 

4a~2i 



30°3 

47l, 



25 



29°34 
5534 



50" 

54*52 



30*3 
54*» 



3I"29 
53*23 



32°3 2 

51 



33 37 
50*57 



34"2j 
50 



35' 



3b'e 

4B*i* 



36*52 

47*6 



37^7 

-46*. 5 



384} 39^ 
4**5 



^9*5 J 40 52 
A 3*2 



26 



31 
53«| 



31*54 
53°8 



32*34 

5 2°;tt 



33 
6(35 



33'58 

50 



344S 

494S 



353. 
49% 



35 "(9 
4 8 



37io 
47* a 



38"a 
46°. 2 



36 56 

45*0 



27 



3I"3 
54% 



31 39 
53*37 



32 S7 
52''9 



333 



34 
50341 



35^ 

49%7| 



35"3 

4945 



55 
48*5i| 



36 3t 

4*\ 



37a5 

A7*7 



40 4 
44''.o| 



42' 



28 



32''2 
53*22 



32°39 
52*39 



33°ie 
5IS6 



3357] 
5 



3439 
50*25 



36^7 
48l, 



36°52 
47^6 



3740 
47 



39\ 



40*(i 



4r» 

43"'9 



29 



30 



31 



32 



33 



34 



35 



36::, 



37 



38 



39 
40 
41 



3259 3338 

52*27 5 



3458 

so"/* 



3539 
49 i9 



36°23 

48*4 



33' 

5(*33 



34'36 
50 50 



35*5 
50*7 



37 8 
4750 



37l^ 
46 



3556 

49*2 o 



37l 



38^7 



36*36 

4834 474S|46"55|46 



e46. 



4f.3 



3653 
3 



394i|4032|4l IS 
45*9 



3+53 

50' 



35^ 
4950 



353 

49°57| 



36i 

49* 3| 



36 
48"»8 



373 
4.739 



3820 
4652 



39 5 
46*1 



3952 

4.5 



4i°at 

43*20 



42.M 



36V 



37<b 
48°2 



374S 
2 47a 



38 3/ 
4.649! 



36*3 9 



39':i5 

4558 



40°/ 
45°» 



40 49 
44' 



4228 



384 2 
46*46 



39"k6 



4056 

44' 



371 



• 43" 



38°. I 

472 7 



46' 



39°a5 

4547 



40^ 
45*8 



41°-^ 



41*49 
5L9 



3a°22 

47I4 



39^ 



39°3 
46*'39 



4241 



39 62 
45*52 



4 2*0 
43*34 



424J 



40*0 

4552 



40"*o] 
45*8 



4/22 



4-2"5 

43*37 



40"47 
45*7 



All, 

44*241 



42"9 

43391 



42''.4 
43*40 



43l°.8 

4342 

43'^ 



PROVIDENCE, R. I. 



27 



TABLE 3. — {Contimied.) 



Angle of Face. 

GEAR. 



2G25 24 23222 



2019 



18 



17 



16 



14 



13 



12 



12 
13 
14 
15 

li 
17 

]^ 
19 



20- 



21 3. 

50*5 



23 6 



Z4 

56*4.9 



Z5\ 
55°3Z 



26*3 

54*7 



5239 



28*25 
5;* 3 



3:"ii 
\A-7l 



32i* 

4-5°24 



3426 

4.3"i* 



3oi 
4Q°A0 



38*r 



22 3T 
S9°29 



23-2 
58 is 



25''s 

S6'r. 



28V 

52 



2! 

50"39 



30 < I 



32V 

47°6 



53 34 3< 

45 22 43°2o|4-[°9 



384a 



24^2 i 



ZS 

5646 



26°8 

5538 



27 5 
54°i5 



284. 



47*6 



34-8 
45^2 



3540 
4-32 



3728 

4 1 "2* 



26' 
5613 



27 3 
55*7 



27"« 
53°58 



Z&'SB 
5244 



ao-o 
5 



3re, 
SO't 



34ff« 
45 



ae'ia 



3747 

4i°39 



2 7% 2 

54' 



28*4.5 
53°3 



29*3| 
52°2 



30 V» 

51*6 



dl'ss 

48°22 



3+°(2 
46*S2 



3531 
k«-5°i9 



36'i4 
43 



38°2J 

4l°5f 



2330 



30a»| 
52 



31 26 

50*8 



32 28 
49°3 2 



3335 

48 



3447 

46' 



at'o 



3 721 

143 a: 



384i 
42° I 



31 5 

514/ 



50*32 



33°* 

4 9° 8 



34 8 

48°2 



45 



374, 



395 



32°36| 
50° B 



3336 

49°8 



34 38 
4754 



36 5 3 

45 



386 
43 50 



3924 



20 



34° 5 

48*17 



35°6 

4746 



36''b 

46°32 



3&26 

43°S2 



39*39 
42°27 



21 



35*3 
4739 



36°32 
46*2 B 



3 737 
/5°3 



3844 

43''56 



39S4 
4i°34 



22 



3652 

46°24J 



375 5 
45°. 3 



4356 



40 e 
4-2°4o 



23 



38 la 
45*2 



4020 

4246 



41 16 



24 



39*29 

44 "3 



403J 

42°ai 



4138 



25 



40 43 
42^7 



26 



g, = 90° - {a, + /5) 
g, = 90° - {a, + /3) 



(Seepage ij.) 



28 



TABLE 4 
Angle of Face. — Gear. 



70l6S|68l67|66|65|64|63|62|6i 



7271 



5/ 8* 
37 77 



60S35857 



12 



7-53 



8" 



7^59 n'SO 



78*39 



78'39 



B'ZI 
78°/9 



8-28 
78'/0 



8' 35' 
77*59 






9'r 



9'n 

77*1' 



76*4» 



9' 35 
76*5i 



-AS 9'5S' 
5•^^ 76' « 



rs 



40 
78 12 



8'48 
78 2 



8*54 
7752 



7742 



9* a' 



3'f8' 
77*20 



9*26 
77* 8 



9'3i 
7656 



9* 4 J S'sz 
76*45 7^32 



10' I 
76*19 



10* II 
76 7 



W2l' 
75S3 



10 31 
75*33 



10*42 10*53: 



75 26 



7S-II 



14 



77 zi 



34 
7716 



9*42 
77*4 



9:50 

76 54 



9*59' 
76*43 



10^ 
76 30 



10* l£ 

76*18 



lO'as 
767 



lo^asiio 

7555 



7541 



10*54 
75*2»' 



iry 

75*15 



11* 16 

75* 



ir27 

7445 



74 



39 ll'W 



l*3l' 74*14 
12*35 l?4« 
73*35 73 18 

13*30 13^ 



IS 



764C 



I0;2l' 

76 Z9 



I0*3I»' 

76 18 



|0>' 
76 6' 



10*47 
75 55 



10:57 

7543 



11*6 
75^30 



iri6 

7516 



75 3 



II* 37 

74*49 



11*41 



11*59 



74 35 74 2» 



12* U 
74*7 



I2.*Z2 

73*56 



16 



lO'ssi 

75*55 



K7'. 
7543 



75 31 



11*26 
75 20 



Jr37 
75 7 



7454 



11° 56 12:7 
74*4*7427 



tt' 17 

74*13 



12*40 12*52 
73447^*28 



13*5 

73*ii' 



13*18 



72 ii 
J4'26i4^ 

71*45 7r2« 



17 



11*44 

75*10 



n:54 

745a 



12*4 

744« 



12*13 
74'33 



12*24 

74*26 



12*34 

74'S' 



12*4^ 
73°52 



I2*S^ 
7338 



13*7' 
7323 



13^ 
73*9 



I3*3t 
7252 



13*45 
72*35 



13*59 
7?*2I 



14' l< 
72 3' 



18 



12*29 



I2*4fl: 



7425 74*12 



12*50 
74* 



f3 

^'46 



73'3Z 



13-23 
7J*I9 



I3'34 

73*4 



IV47 

7243 



l3*5Sf 
7233 



14*24 
72Z 



14*38 
7144 



14*52 
71*28 



15*6 
71*10 



7D*5i' 78i4 

69'S9 63*85 

17*8' Wat 

69*6' 68*46 
18* 2' 1 18*30 



Id 



13*14 

73*40 



13:25 
7327 



»:36 

73 14 



1^48 

73* 



14* 
72*46 



7t3i 



WZ4 
72*16 



14*36 
72 



I4*4i 
71*45 



28 

i?5? 

7838 



15*15' 

71*11 



15*56 15*44 
705470 30 



I5;s9 

70 ri 



20 
21 



13*59 
72*57 



14; II' 
7243 



14:23 
7^29 



14:34 
7214 



14*46 
72^ 



15*4' 
7r45 



15*11' 
71*29 



15*25 

7ri3 



I5»39 
7056 



16*7 
7021 



le'ei 
70*3 



J6*.37 
6945 



16*53 

69*2S 



14*43 
72*13 



K'SS 
71*59 



15* a' 
7/44 



15'ai 
7/*29 



15*33 

7/*ia'' 



is;a6 

7058 



i5;ss( 
7041' 



J6*83 
70*25 



16*58 

70*8 



69' 



16*58 
69*35!; 



I7*?ll 
69*11 



IT*28 I7°46 
68*54 6**34 6%" 



14 67 52 
18° 56; 19* li 
67'2£ 67° I 
I9"4a:20^ 



22 



15*27 15*4<> 



15*53' 



7129 



7114 70 59 



16**; 

70*44 



e6°20 
70*28 



16*33 
70*11 



16*47 



17*2' 



6955 69 38 



17*16' 
69*20 



17*31 
69*1 



17*49 
68*4^' 



16*3 
68*2) 



18*20 

68*4' 



18*37 
67*43 



17*50 
68*so' 



23 



70*46 



16*24 
70'30 



70 16 



16*51' 
69*59 



17; 5' 
6943' 



I7;2b 
692669 



17*34 
8 



18*5' 
68*33 



18*20' 



68*14 67*54 



18*54 
67*34 



19* 10 
67*14^ 



119^ 
66*52 



66*32 eeTgr 

20^ «»^ 
6^41 65*<8 
21*29 2?^ 

64*51 64 a 

2?Sq22^ 
64*2 63*39 
23*10 23*31 



24 



16*55' 

3' 



17*9 

69*47 



17*22 
69r3z 



17*37 
69*15 



17*51' 
68*59 



18*6 
68*48 



18*21 
68*23 



18*37 
5 



IV53 
67*45 



19*9' 
67*27 



19*26 
67*6^ 



44 28* 



19* 
66*46 



6625 



20*19 
66*3' 



25 



I7*39|l7*5i 
69*21 



18*6' 
68*48 



18*21 

68*31 



18*36' 
68*14 



18*52 
6756 



19*7 
67*37 



19:2^ 19 



67/8 



67 



19*57 
66*39 



2^14 20* «L 



6620 



65 58 



20*51 
65*37' 



21*10 
65*14 



26 



1^2 

68*39 



18*36 
68*22 



18*51 
68*5 



19*6 
67*48' 



19*22' 
67*?d 



19 57 
67*13 



19* Sa 
66*53 



20"10 
66*34 



20*26: 
66*14 



6S*53 



21*2 
6$*32 



21*21 

65*11 



21*41 
64*49 



20*38' 
66*8' 



22' 
6426 



27 



19' 3' 
67*57 



19*19 
67*39' 



19*34 
6/22 



19*49 
67*5 



20* «■ 
66*46 



20*2£ 
66*28 



20*56 
6S48 



21*13' 
65*29 



21*3^ 
6S*8' 



21*50' 
64*46 



10 22" 



24 64* 



22*49 
63*38' 



63*14 62*49 

2arsa'a4^ 

62*27 62*1' 
24*48' 28*10 
6I*4« 61*14 
25^3^ 
60*54 60*27 



28 



»*4«' 
67*16 



20*1' 
66*59 



20^ 
66*41' 



20*32 
66*22 



20f>50 
66' 



21*6 

65*44 



21*2^ 
6^25 



21*41 
65*5' 



22 

64*44 



22*18' 
64*^22 



64* r 



22*56' 
69*38' 



2^7 
63*15' 



23*38 
62*52 



29 



2flr27 

66*35 



2043 
66*17 



20*59' 21*1^ 
65*59 65*40 



2I<'33' 
65*21 



21*50 
65*2' 



22-8 
64*42 



64*21 



22*45 
63*59 



23* 5' 

6^37 



23r25 

63*15 



2^4^ 

62*52 



24-*4 
62*28 



24*2i 
62*s 



30 



21*9 



21*25 



65*55 6!r37' 



2lW 
65*18' 



21*58' 

6458 



22^ 
64*39 



22*34 22" 



6418 



52 
63' S8 



23*«tf 
63*38 



23*3d 

esTie' 



23^ 

62*54 



24*10' 

62* ad 



a4'M' 

62*7 



24*51' 
6I'45' 



25*12 
6i*l8' 



24*14' 24*34 
62*10 



31 



21*50' 
6S*I4 



22*6' 
6456 



22*2^ 
64*36 



64*17 



2259 
63*57 



2i*n 

63*37 



23*35 

63*1!; 



23*5^ 

62* 



55' Si^^ 



24*54 
61*46' 



25*17 
61*23 



25*3S 

60*58' 



25*8? 26' 



60 34 



68' 8' 5342 
27*9' 2ri4 
59*23 58*56 
2fi?2fi9 
58*38 58*11 
28*4<<29*? 
ST54 57*27 
29«2?« 



32 



6435 



22*48' 



23*4 

6ys6' 



23*23 



23*40 

63*.I6' 



2^59' 
62*55 



2418 
62*34 



6^12 



24P58' 
61*50' 



25*18' 
61*26 



2r3si 

61*3' 



60*3^ 



26*23' 26*45^ 
60* l!> 59*49 



33 S- 



23*10' 



2r28' 
63*36 



23*46 
63*16 



24?4' 
62* 



Z^zi 24*41 



ZS"! 



56' 62*36 



62 15 6I'S3 



2^21 
6I*3»' 



25*42 
61*8 



26-^ 
I' 68*44 



26;24 
60*20 



2F4s: 

55 



27*9' 
59*31' 



27*31 
59' 5' 



34 



6317 



24*8 

62*sil 



24-*27 
62*37 



24*4425 



62*16 



4 
61*56 



25*23 
61*33 



25*42 26*3 



61 12 



25^ 
6I-I6' 



6049 



t26°4^ 
60*2 



27*7' 
59*37 



27*29 
59*13 



2fs2 
58*48' 



28*16 
S8*22 



35 



24*29 
6f39 



24*48' 
6218' 



25*6' 

61*58 



25*25 

61*37 



26:4 
6054 



26*24 
6032' 



26*45 
60*9' 



27*6 



27Vi 



59*44 59* 



tr^ 



27*50 
58*56 



28*13 
58*31' 



S7S0 



28*36' 
58*6 



5724 



29- 1 
57*39 



5711 



5644 



36 



25*9' 
62* i: 



25*27 

61*41 



25-45 
61*20 



26*5 
60*59 



26*24 
60*36 



26*45' 
60*15 



27:5 
5951 



2r26' 
5928 



594 



28*10 
;58" 



40 58 



28:33 
15 



28*56 29' 



5657 



30*9' 30^ 
56*29 56*1'. 
3015 3Pli 
5548 552 
3l*3rf3ri 



37 



25*47 
61*" 



26*6' 
61* 



60*47 



26*25 
60*41 



26*44 
60*20 



27>' 
5958 



2^ 
5935 



27*45 
59*13' 



28*7 
58*49 



25 



28*51' 
58 



^9^ 
57*35 



23rM 30*2 
57 lb 56*42 



38*27 
56*15 



38 



26*44 
60*26 



27:4 
604' 



5942 5920 



28;4; 
58*56 



5834 



28*4-7 
58*9' 



29* 



5745 



►' 29:3^ 
5 57*21' 



29*5i 
56*55' 



30*20 

56*30 



30*44 
55*2' 



31*9' 
5^35 



55 7' 5439 
3?l6 32*4< 
54 28 53 51 
32*57 33*24 
53*46 5318 
33* 3i 34*1 
53* 7' 5238 
A9 34V 



33 



27*3' 

60*9' 



27*22 

594^ 



27/42 

59 28 



2»*2' 
59*4' 



28 22' 
58*42 



28;4^ 
58*19 



29*5 
5755 



29*27 
57*31 



29^ 
57*7' 



30*13 
56*41 



30*36 31*1' 



56 16 



5549 



31* 2i 

5522 



31*50 
54*54 



27*4<< 
59*34 



2n»3 

5932 



40 
4i 
42 



28'2rf 

58*5«i 



28*41 
58*27 



29*1' 
58*5 



57*42 57* 



29*43' 
17' 



30*6' 
56*54 



30*52 
56 2' 



31* 17 
55*37 



31*42 
55*16 



32*6' 

54*44 



32;3i 
54*15' 



2rir 

58*57 



2^37 
5837 



28*«' 

58*22 



29'ia; 
58*1 



28*57 
IS' 



29*18 
57*52 



29*39' 
57*29' 



3crT 

57* 



3Q*22 



4 5640 



30*45 
56*15 



31* 9' 3l*si' 
55* St' 55* 25 



'4r3rior 

13* 54*48 



31*55 

54*59 



3?io!??vi 



5432 



54 4' 



33*12 
53*36 



29' 
57 39 



33 29' 



55 

57" IS* 



30*16 
56*52 



30*38 

5^28 



31* 

56*4' 



31*23 
55*39 



32*35 

54*21' 



33' 

53*54 



3^261 
53*26 



33* 
52*58 



^34? 



52 29 52 



TABLE 4. — (Contimted.) 
Angle of Face. — Gear. 



29 



56 55 S4 53 52 51 



504948474645444342 



12 



75 SA 



10* le' 

7540 



lO'.ZB 
7524 



75 9' 



10* sz 
74 si 



74 $7 



7415 



ir 30 

7358 



11*43 

73*39 



II* «' 
73° 20 



It" m 

7i5B 



1755 

72 37 



I2*4J^ 
72 15 



irr 

71*51 



13*19 
71*25 



j3 
14 



7456 



1116 

744« 



h;28 

7424J 



IfAZ 
74 8 



11*54: 
73*50 



12*8 
73*32 



IZ'zd 
73° IZ 



12*37 
7253 



12; 51 
72*33 



13' 7 
72 n' 



13* Z3' 
71*49 



1340 
71*26 



I3*S« 
71-2' 



K 

70 38 



14*35 
70"li 



KTz' irib 
73*58 7342 



12*2912' 



7325 



43 
73 7 



12*57 
7i49 



1 3* 2B 13*43' 



72*29 



»3*26 
728 



71 A^ 



71*17 



71-5 



33 

70'4» 



I4*si 
l(fl7 



15" 10 



IS'36 



69 si 69*26 



15'SI' 
68*59 



15 



13' 1 
7/1' 



I3*I6" 
72*44 



59 



A-i 



9 14' »4 
7r2fl 



14*30 
71*6' 



14*47 
70*45 



15*5 
7tf23 



15^3 

69*59 



15*42 
69*34 



16' r 

69 9 



16*22 
6&42 



16*43 

6iiS 



67*47 



J6 
17 



•3*59 
72*5' 



14*13 
71*47 



14*28 
71*2%' 



I4*44' 
71*8' 



15" I' 
70*49 



7027 



l6i 



1^52 

69*42 



16* u' 
69*19' 



16*30 

68*54 



50 



17*10 
68*2* 



I7*3t 

67*34 



irsei 

67*6" 



18* »• 
66*36 



I4°57 
71*9' 



15-u' 
70*49 



I5*2«i 
70*30 



15*44 
7016 



16* r 

69*49 



»*I8 
69*26 



16° 37 
69*3' 



16*55 



17*15 



68 39 68 15 



17^36 I7;57' 
67*50 67*23' 



18*20 

66*54 



I8*4» 
66*27 



19*6' 
6S*S8 



19^31 
6^27 



19 



I5°5i 
7014 



16*7 
69*53 



16*26 
6934 



16*42 
69*12 



17' r 

68*43 



17*20 

26 



17*39' 

68*3 



n*58 

67*38 



18*20 

67*12 



18*44' 
66*47 



19*3' 
66* 19 



19*27 

65*si 



19*50 
65*20 



20 18 
64*5« 



2tf« 
64*18 



I6*4«f 
69*19 



17*2 

68*58 



»7*23 



17*41 



68*37 68*15 



18* 
67*52 



18*21 
67*29 



18*40 

67*4 



19*1 
66*37 



19*22 
66*12 



i9*4c; 

65*44 



20° 8' 
65*16 



20*34 
64*44 



20*59 
IS 



21*24 
63*44 



21*52 
63*10' 



20 



17*44 
68*26 



18" I' 18* 19' 
6«r3' 6/41' 



18*40' 
67*18 



19*20 



19*41' 



66*54 66*30 h€ 5 



20*2' 
65*38 



20*25 
65*11' 



20*43 
64*43 



21* 13' 
64* li 



21*33 
63*43 



227 
63*11 



22*32 

6^38 



23' 

162*4" 



21 



18*39 
6l*3r 



18*57 
67*9' 



19*16' 



19*37 



664666 23 



19*58 
6S*S8 



20*19 
65*33 



20*41 
65*7 



21* y 

64^39164 



21*27 
11' 



21*52 
63*42 



22*17 
63*13' 



2?4S 

6241 



23*|d 
6£8' 



23*38 

61*34 



2<8' 
61 



22 



19*32' 
66 38 



2^ 
6616 65*52 



I9*5i 



20*33 
65*27 



20'S5 
6^3' 



21*17 
6437 



22r 13 
634t 



21*40 
6410 



22*5' 
63*41 



22*ri 
63*13 



22*53 
6243 



2319 
62* ii' 



23*46 
61*40 



24 
61*7' 



24*44*8*14 



60*3; 



S9S6 



23 



zo'zi 

65*47 



20*4i 
65 23 



21*8 
6458 



21*29' 
6433 



2r52' 
64 8' 



22*37 
6513 



2af2' 
62*44 



23' 27 
62*15 



23*54 

61*44 



24*21 
61* 13 



25*2» 
60* IS' 



24^49 
6044 



25*18' 
60*6 



59*3l' 



26*18 
58*54 



24 



21* 19 
64*55 



21*39 

64* ai 



22* r 

64*5' 



2r24 

63* 



22*46 23jtf 



63 14 



6246 



23*36 
62* 19 



24* 
6I*4«' 



24*26 
61*18 



24*53 

60*47 



2sr4a 

5941 



IZ^Zi 
59 6 



26*51 
58*31' 



27;23 
57*53' 



25 



22*11' 



22*33' 
6339 



2fS6' 
63*14 



23*18' 
62*48 



2^ 
62*^ 



24*13 
61*56 



23°4t' 
62*21 



24*7 
61*53* 



24*3i 
61*24 



24*57 
60*53 



25*24 
60*22 



25*52 
59*50 



2^ 
58*55 



26*20 
59*18 



26'50 
58*44 



27*21' 
Si'd' 



2r52 

57*32 



28*26 

56*S4 



26 



23*3' 
63^5 



23*25 
6249 



6128 



25" I 
60*59 



25*28' 
6030 



25*53 

59 59 



26*21 
27 



27*19' 
58*21 



27S»^ 
57*47 



28*2i 
S/ll 



W 



2927 



56*34 SS*5S 



27 



2F5? 

62* 



2S6l 



24> 
58 



2440 25' 



6152 



1*5' 



2?5? 
6043 



25-29 
60*37 



25'55 
60*7 



26*22 

59*38 



26*48' 
59*5' 



27*17 
58*33 



27*46 
SB- 



57*8 



28*16; 
57*26 



28*4f 
56* 5l' 



29*19 
56* IS 



29*si 
55*38 



30^ 
54*42 



lcr«7 

54*59 



5r25 
54* 



2| 

29 



24*441 
61*36 



61*9' 



25*56' 
6014 



59*46 



26*4« 

59*16 



27*15 
58*45 



27*43' 
58*13 



28*1? 
57*42 



29* li 
56* si 



^9^^^| 

55*5^ 



3flric; 
55*2* 



25*33' 25*57 
60*47 60*21 



26*22 

si 



26*47 
59*25 



2ri4 
58*56 



27*40 
58*26 



28*8 
57*5^ 



28* 
5/22 



29*5 

S649 



29^ 
5^15 



30*8' 
55*40 



5i'24 544<i 



30*40 
55*4 



3p3? 
54* •« 



31*13' 

54*27 



31*48 
53*48' 



32?3 

53*9 



32"44j83*l9 
5254 52*1^ 



30 



26* 2t 
60* 



26*47 
5933 



59*6 



27 

58*36 



2iV 
58*6' 



2832 
5736 



29' 



29*S9 
55*58 



3PSS 
53*57 



51*23 



31 



2ri6| 
59*14 



27*34 
S8*46' 



283' 

58*15' 



28*2/ 
S7*4e' 



28*54 
57 18 



29*2J 
56*47 



29*sr 
56*15 



30*2i 30°52 



5542 



55 8 



3r22 

5434 



32'29 

53* 2t 



33*2 

S2*42 



33*39 

52*3 



32 



27*58 
58*28' 



28*23 
57*59 



28°4«l 
57*31' 



2Sri7 
57*1 



29*33' 

56 41' 



56* 



30*42 
5^28 



31*10 
54*54 



31*4^ 
54*20 



32*14 32*46 



3?3e 
53*32 



5344 



53 8' 



33*21 
52*31' 



3r56 

51*52 



34*31' 
51*13 



35* 8 

50*32 



33 



57*43 



29^(0 
57*14 



56*45 



30*5 
S^IS 



30*32 
55*49 



31* 
55*13' 



3r3\ 
54*39 



32ri 
54*5 



,93:4 



33;3af 

52 20 



34*12 
51*41 



34*47 
51*3' 



35*24 

SOW 



36' 
49*41 



34 



23*31 
56* 



59 56' 



29*57 
'29 



30*24 
§1 



30*5t' 
55*29 



31*20 

54*5i 



31*49 
54*27 



32*19 
53*53; 



32*5i 
53*20 



33*22 



3^3?ii 



5^44 52 8' 



5131 



SCTso 



50 14 



49*3i 



36*51' 

4853 



35 



lOM? 
5^15 



30*42 
55*46 



31* Itf 31*38 



55* 16' 



5444 



3^7' 
5^13' 



32*3^ 

5340 



33'7 
53* 



SPi8 
5234 



34^334*42 
51*58' 51*22 



35*17 

50*45 



35*51 
50*7 



36*27 
49 27 



37*5' 
48*47 



3ir42 
48 6' 



38^ 

47*20 



36 



31 

55*32 



31*27 
55*3 



31*55 
5433 



3f23 

54*1' 



3^53 
5^28 



52 57 



33*53 
52 23' 



34*25 



34*57 



5l'4« 51* 13 



35*31 36*5' 
50 37 4959 



36*41 
4921 



37 16 

48*4i 



97*53 
48*1 



37 



34*45' 
544) 



32*i«t 32*40 
54 20 53 so 



33*8 
53*18 



33'38 

S2*4€ 



34*9' 
5213 



34"4«' 
51*40 



35*12 35*4^ 



51*^ 



50*29 



36*18 36*51 
4952 4915 



37^ 38*4' 



4837 



4756 



3842 ^.20 
47*16 4634 



38 
39 



54 9' 



3?S6 

53 



3r24 

5^8 



33*52 
52*38 



34*22 
52*4 



34*54 
51*30 



35*24 
5056 



35*5- 
50 2 



3er29 
4945 



37*3 
43*9 



37*3« 
48* 3i 



38*14 
47*52 



3r5r 

47* 13 



39*28 
46*32 



40*7 
43*51 



401545S 



33*10' 
53 28 



33* 

si Si 



3^34' 



7 
5227 



34* 
51*54 



3(95' 



.°7 

51*2 1'l 



»5*37 

50*49 



36*9' 



36*41 Sri? 
49*39 49*3 



37*48 
48 



26 47 



38*24 



39* 

47*16 



39r36 

46*30 



4545 



i^ 



40 
41 



33*52 
52*46' 



34*21' 
52*17 



34*50 
51*46 



35*18' 
51*14! 



35*49 

S0*4« 



36*20 
50*8' 



36*5»' 

49'3i 



37*2i 



48*22 



38*33 

47*45 



39*8 
47*6' 



39r44 40*20 
4628 45*48 



4tf58 



4I*»7 



45*6' 4425 



S4?3? 
5^9 



35*3' 
51*37 



35*31 36* I 
51* 7' 50*33 



36*31 
50*1 



37*3' 
49 27 



37*3^ 
48*53 



3V7 
40*17 



38*4< 
47*44' 



39' lb 
4f4 



39*51' 



40*2? 4<* 5 
45*7' 



44*4* 4^21 
43*44 



4426 



42 



3ri4 35 



51*30 



*4i' 

50*5!( 



36*12 
50*28 



36-42 
49*S4l 



37*13' 
4^2«' 



37*44' 
48*48 



3tfl7 
48*13' 



38*49 
47*37 



39*23 

47*1 



39'S8 
46*24 



40*34 
45*46 



41*9 
45*7 



41*41 
+4£7 



Aiti 
43*4* 



«4 



30 



BROWN & SHARPE MFG. CO. 



NATURAL SINE. 



Deg. 


0' 


10' 


20' 


30' 


40' 


50' 


60' 







.00000 


.00291 


.00581 


.00872 


.01163 


.01454 


.01745 


89 


1 


.01745 


.02036 


.02326 


.02617 


.02908 


.03199 


.03489 


88 


2 


.03489 


.03780 


.04071 


.04361 


.04652 


.04943 


.05233 


87 


3 


.05233 


.05524 


.05814 


.06104 


.06395 


.06685 


.06975 


86 


4 


.06975 


.07265 


.07555 


.07845 


.08135 


.08425 


.08715 


85 


5 


.08715 


.09005 


.09295 


.09584 


.09874 


.10163 


.10452 


84 


6 


.10452 


.10742 


.11031 


.11320 


.11609 


.11898 


.12186 


83 


7 


.12186 


.12475 


.12764 


.13052 


.13341 


. 13629 


.13917 


82 


8 


.13917 


.14205 


.14493 


.14780 


.15068 


.15356 


.15643 


81 


9 


. 15643 


.15930 


.16217 


.16504 


.16791 


.17078 


.17364 


80 


10 


.17364 


.17651 


.17937 


.18223 


.18509 


.18795 


.19080 


79 


11 


.19080 


.19366 


.19651 


.19936 


.20221 


.20506 


.20791 


78 


12 


.20791 


.21075 


.21359 


.21644 


.21927 


.22211 


.22495 


77 


13 


.22495 


.22778 


.23061 


.23344 


.23627 


.23909 


.24192 


76 


14 


.24192 


.24474 


.24756 


.25038 


.25319 


.25600 


.25881 


75 


15 


.25881 


.26162 


.26443 


.26723 


.27004 


.27284 


.27563 


74 


16 


.27563 


.27843 


.28122 


.28401 


.28680 


.28958 


.29237 


73 


17 


.29237 


.29515 


.29793 


.30070 


.30347 


.30624 


.30901 


72 


18 


.30901 


.31178 


.31454 


.31730 


.32006 


.32281 


.32556 


71 


19 


.32556 


.32831 


.33106 


.33380 


.33654 


.33928 


.34202 


70 


20 


.34202 


.34475 


.34748 


.35020 


.35293 


.35565 


.35836 


69 


21 


.35836 


.36108 


.36379 


.36650 


.36920 


.37190 


.37460 


68 


22 


.37460 


.37730 


.37999 


.38268 


.38536 


.38805 


.39073 


67 


23 


.39073 


.39340 


.39607 


.39874 


.40141 


.40407 


.40673 


66 


24 


.40673 


.40939 


.41204 


.41469 


.41733 


.41998 


.42261 


65 


25 


.42261 


.42525 


.42788 


.43051 


.43313 


.43575 


.43837 


64 


26 


.43837 


.44098 


.44359 


.44619 


.44879 


.45139 


.45399 


63 


27 


.45399 


.45658 


.45916 


.46174 


.46432 


.46690 


.46947 


62 


28 


.46947 


.47203 


.47460 


.47715 


.47971 


.48226 


.48481 


61 


29 


.48481 


.48735 


.48989 


.49242 


.49495 


.49747 


.50000 


60 


30 


.50000 


.50251 


.50503 


.50753 


.51004 


.51254 


.51503 


59 


31 


.51503 


.51752 


.52001 


.52249 


.52497 


.52745 


.52991 


58 


32 


.52991 


.53238 


.53484 


.53730 


.53975 


.54219 


.54463 


57 


33 


.54463 


.54707 


.54950 


.55193 


.55436 


.55677 


.55919 


56 


34 


.55919 


.56160 


.56400 


.56640 


.56880 


.57119 


.57357 


55 


35 


.57357 


.57595 


.57833 


.58070 


.58306 


.58542 


.58778 


54 


36 


.58778 


.59013 


.59248 


.59482 


.59715 


.59948 


.60181 


53 


37 


.60181 


.60413 


.60645 


.60876 


.61106 


.61336 


.61566 


52 


88 


.61566 


.61795 


.62023 


.62251 


.62478 


.62705 


.62932 


51 


39 


.62932 


.63157 


.63383 


.63607 


.63832 


.64055 


.64278 


I 50 


40 


.64278 


.64501 


.64723 


.64944 


.65165 


.65386 


.65605 


'49 


41 


.65605 


.05825 


.66043 


.66262 


.66479 


.66696 


.66913 


i 48 


42 


.66913 


.07128 


.67344 


.67559 


.67773 


.67986 


.68199 


47 


43 


.08199 


.68412 


.68624 


.68835 


.69046 


.69256 


.69465 


40 


44 


.69465 


.69674 


.69883 


.70090 


.70298 


.70504 


.70710 


45 




60' 


50' 


40' 


30' 


20' 


10' 


C 


Deg. 



NATURAL COSINE. 



PROVIDENCE, R. I. 



31 



NATURAL SINE. 



Deg. 


0' 


10' 


20' 


30' 


40' 


50' 


60' 




45 


.70710 


.70916 


.71120 


.71825 


.71528 


.71731 


.71984 


44 


46 


.71934 


.72135 


.72836 


.72587 


.72787 


.72930 


.78135 


43 


47 


.73135 


.73333 


. 73530 


.78727 


.78928 


.74119 


.74314 


42 


48 


.74314 


.74508 


.74702 


.74895 


.75088 


.75279 


.75471 


41 


49 


.75471 


.75661 


.75851 


.76040 


.76229 


.76417 


.76604 


40 


50 


.76604 


.76791 


.76977 


.77102 


.77347 


. 77581 


.77714 


39 


51 


.77714 


.77897 


.78079 


.78260 


.78441 


.78621 


.78801 


38 


53 


.78801 


.78979 


.79157 


.79335 


.79512 


.79688 


. 79863 


37 


53 


.79863 


.80038 


.80212 


.80385 


.80558 


.80780 


.80901 


36 


54 


.80901 


.81072 


.81242 


.81411 


.81580 


.81748 


.81915 


35 


55 


.81915 


.82081 


.82247 


.82412 


.82577 


.82740 


.83903 


34 


56 


.82903 


.83066 


.83227 


.83388 


.88548 


.88708 


.83867 


88 


57 


.83867 


.84025 


.84182 


.84389 


.84495 


.84650 


.84804 


32 


58 


.84804 


.84958 


.85111 


.85364 


.85415 


.85566 


.85716 


31 


59 


.85716 


.85866 


.86014 


.86162 


.86310 


.86456 


.86603 


30 


60 


.86602 


.86747 


.86893 


.87035 


.87178 


.87820 


.87463 


29 


61 


.87462 


.87602 


.87742 


.87881 


.88020 


.88157 


.88394 


28 


63 


.88394 


.88480 


.88566 


.88701 


. 88885 


.88968 


.89100 


27 


63 


.89100 


.89282 


.89863 


.89498 


.89622 


.89751 


.89879 


26 


64 


.89879 


.90006 


.90182 


.90258 


.90888 


.90507 


.90680 


25 


65 


.90630 


.90753 


.90875 


.90996 


.91116 


.91285 


.91854 


24 


66 


.91354 


.91472 


.91589 


.91706 


.91821 


.91936 


.93050 


28 


67 


.93050 


.92163 


.92276 


.93388 


.92498 


.92609 


.92718 


22 


68 


.93718 


.92827 


.92984 


.98041 


.93148 


.93253 


.98358 


21 


69 


.93358 


.98461 


.93565 


.93667 


.93768 


.93869 


.98969 


20 


70 


.93969 


.94068 


.94166 


.94364 


.94860 


.94456 


.94551 


19 


71 


.94551 


.94646 


.94739 


.94882 


.94924 


.95015 


.95105 


18 


73 


.95105 


.95195 


.95288 


.95871 


.95458 


.95545 


.95630 


17 


78 


.95630 


.95715 


.95799 


.95882 


.95964 


.96045 


.96136 


16 


74 


.96136 


.96205 


.96284 


.96868 


.96440 


.96516 


.96592 


15 


75 


.96593 


.96667 


.96741 


.96814 


.96887 


.96958 


.97029 


14 


76 


.97039 


.97099 


.97168 


.97237 


.97304 


.97371 


.97437 


13 


77 


.97437 


.97502 


.97566 


.97629 


.97692 


.97758 


.97814 


12 


78 


.97814 


.97874 


.97984 


.97992 


.98050 


.98106 


.98162 


11 


79 


.98162 


.98217 


.98373 


.98325 


.98378 


.98429 


.98480 


10 


80 


.98480 


.98530 


.98580 


.98628 


.98676 


. 98723 


.98768 


9 


81 


.98768 


.98818 


.98858 


.98901 


.98944 


.98985 


.99036 


8 


82 


.99026 


.99066 


.09106 


.99144 


.99182 


.99318 


.99254 


7 


83 


.99254 


.99289 


.99328 


.99357 


.99389 


.99431 


.99452 


6 


84 


.99452 


.99482 


.99511 


.99539 


.99567 


.99593 


.99619 


5 


85 


.99619 


.99644 


.99668 


.99691 


.99714 


.99785 


.99756 


4 


86 


.99756 


.99776 


.99795 


99813 


.99830 


.99847 


.99863 


3 


87 


.99863 


.99877 


.99891 


.99904 


.99917 


.99938 


.99939 


2 


88 


.99939 


.99948 


.99957 


.99965 


.99972 


.99979 


.99984 


1 


89 


.99984 


.99989 


.99993 


.99996 


.99998 


.99999 


1.0000 





- 


60' 


50' 


40' 


30' 


20' 


10' 


0' 


Deg. 



NATURAL COSINE. 



32 



BKOWN &, bHARPE MFG. CO. 



NATUKAL TANGENT. 



Deg. 


0' 


10' 


20' 


30' 


40' 


50' 


60' 







.00000 


.00290 


.00581 


.00872 


.01163 


.01454 


.01745 


89 


1 


.01745 


.02036 


.02327 


.02618 


.02909 


.03200 


.03492 


88 


2 


.03492 


.03783 


.04074 


.04366 


.04657 


.04949 


.05240 


87 


3 


.05240 


.05532 


.05824 


.06116 


.06408 


.06700 


.06992 


86 


4 


.06992 


.07285 


.07577 


.07870 


.08162 


.08455 


.08748 


85 


5 


.08748 


.09042 


.09335 


.09628 


.09922 


.10216 


.10510 


84 


6 


.10510 


.10804 


.11099 


.11393 


.11688 


.11983 


.12278 


83 


7 


.12278 


.12573 


.12869 


.13165 


.13461 


. 13757 


.14054 


82 


8 


.14054 


.14350 


.14647 


.14945 


.15242 


.15540 


.15838 


81 


9 


.15838 


.16136 


.16435 


.16734 


. 17033 


.17332 


.17632 


80 


10 


.17632 


.17932 


.18233 


.18533 


.18834 


.19136 


.19438 


79 


11 


.19438 


. 19740 


.20042 


.20345 


.20648 


.20951 


.21255 


78 


12 


.21255 


.21559 


.21864 


.22169 


.22474 


.22780 


.23086 


77 


13 


.23086 


.23393 


.23700 


.24007 


.24315 


.24624 


.24932 


76 


14 


.24932 


.25242 


.25551 


.25861 


.26172 


.26483 


.26794 


75 


15 


.26794 


.27106 


.27419 


.27732 


.28046 


.28360 


.28674 


74 


16 


.28674 


.28989 


.29305 


.29621 


.29938 


.30255 


.30573 


73 


17 


.30573 


.30891 


.31210 


.31529 


.31850 


.32170 


.32492 


72 


18 


.32492 


.32813 


.33136 


.33459 


.33783 


.34107 


.34432 


71 


19 


.34432 


.34758 


.35084 


.35411 


.35739 


.36067 


.36397 


70 


20 


.36397 


.36726 


.37057 


.37388 


.37720 


.38053 


.38386 


69 


21 


.38386 


.38720 


.39055 


.39391 


.39727 


.40064 


.40402 


68 


22 


.40402 


.40741 


.41080 


.41421 


.41762 


.42104 


.42447 


67 


23 


.42447 


.42791 


.43135 


.43481 


.43827 


.44174 


.44522 


66 


24 


.44522 


.44871 


.45221 


.45572 


.45924 


.46277 


.46630 


65 


25 


.46630 


.46985 


.47341 


.47697 


.48055 


.48413 


.48773 


64 


26 


.48773 


.49133 


.49495 


.49858 


.50221 


.50586 


.50952 


63 


27 


.50952 


.51319 


.51687 


.52056 


.52427 


.52798 


.53170 


62 


28 i 


.53170 


.53544 


.53919 


.54295 


.54672 


.55051 


.55430 


61 


29 


.55480 


.55811 


.56193 


.56577 


.56961 


.57347 


.57735 


60 


30 


.57735 


.58123 


.58513 


.58904 


.59297 


.59690 


.60086 


59 i 


31 


.60086 


.60482 


.60880 


.61280 


.61680 


.62083 


.62486 


58 1 


32 


.62486 


.62892 


.63298 


.63707 


.64116 


.64528 


.64940 


57 


33 


.64940 


.65355 


.65771 


.66188 


.66607 


.67028 


.67450 


56 


34 


.67450 


.67874 


.68300 


.68728 


.69157 


.69588 


.70020 


55 


35 


.70020 


.70455 


.70891 


.71329 


.71769 


.72210 


.72654 


54 


36 


.72654 


.73099 


. 73546 


.73996 


.74447 


.74900 


.75355 


53 


37 


.75355 


.75812 


.76271 


.76732 


.77195 


.77661 


.78128 


52 


38 


.78128 


.78598 


.79069 


.79543 


.80019 


.80497 


80978 


51 


39 


.80978 


.81461 


.81946 


.82433 


.82923 


.83415 


.83910 


50 


40 


.83910 


.84406 


.84906 


.85408 


.85912 


.86419 


.86928 


49 


41 


.86928 


.87440 


.87955 


.88472 


.88992 


.89515 


.90040 


48 


42 


.90040 


.90568 


.91099 


.91633 


.92169 


.92709 


.93251 


47 


43 


.93251 


.93796 


.94345 


.94896 


.95450 


.96008 


.96568 


46 


44 


.96568 


.97132 


.97699 


.98269 


.98843 


.99419 


1.0000 


45 


- 


60' 


50' 


40' 


30' 


20' 


10' 


0' 


Beg. 



NATURAL COTANGENT. 



PROVIDENCE, R. I. 



33 



NATURAL TANGENT. 



Deg. 


0' 


10' 


20' 


30' 


40' 


50' 


60 




45 


1.0000 


1.0058 


1.0117 


1.0176 


1.0335 


1.0395 


1.0355 


44 


46 


1.0355 


1.0415 


1.0476 


1.0537 


1.0599 


1.0661 


1.0733 


43 


47 


1.0723 


1.0786 


1.0849 


1.0913 


1.0977 


1.1041 


1.1106 


42 


48 


1.1106 


1.1171 


1.1336 


1.1302 


1 . 1369 


1.1436 


1.1503 


41 


49 


1.1503 


1.1571 


1 . 1639 


1.1708 


1.1777 


1.1847 


1.1917 


40 


50 


1.1917 


1.1988 


1.3059 


1.2131 


1.3303 


1.3375 


1 3349 


39 


51 


1.2349 


1.3433 


1.3496 


1.2571 


1.3647 


1.2723 


1.3799 


38 


53 


1.3799 


1.3876 


1.3954 


1.3032 


1.3111 


1.3190 


1.3270 


37 


53 


1.3370 


1.3351 


1.3432 


1.3514 


1.3596 


1.3680 


1.3763 


36 


54 


1.3763 


1.3848 


1.3933 


1.4019 


1.4106 


1.4193 


1.4281 


35 


55 


1.4281 


1.4370 


1.4459 


1.4550 


1.4641 


1.4733 


1.4825 


34 


56 


1.4825 


1.4919 


1.5013 


1.5108 


1.5304 


1.5301 


1.5398 


33 


57 


1.5398 


1.5497 


1.5596 


1.5696 


1.5798 


1.5900 


1.6003 


32 


58 


1.6003 


1.6107 


1.6212 


1.6318 


1.6435 


1.6533 


1.6642 


31 


59 


1.6643 


1.6753 


1.6864 


1.6976 


1.7090 


1.7304 


1.7320 


30 


60 


1.7330 


1.7437 


1.7555 


1 . 7674 


1.7795 


1.7917 


1.8040 


29 


61 


1.8040 


1.8164 


1.8390 


1.8417 


1.8546 


1.8676 


1.8807 


38 


62 


1.8807 


1.8940 


1.9074 


1.9209 


1.9347 


1.9485 


1.9626 


37 


63 


1.9626 


1.9768 


1.9911 


3.0056 


3.0303 


2.0352 


2.0503 


26 


64 


2.0503 


3.0655 


3.0809 


3.0965 


3.1123 


2.1383 


2.1445 


25 


65 


3.1445 


3.1609 


3.1774 


3.1943 


3.3113 


3.3385 


3.3460 


24 


66 


2.2460 


3.3637 


3.3816 


3.3998 


3.3183 


3.3369 


3.3558 


23 


67 


2.3558 


3.3750 


3.3944 


3.4143 


3.4343 


3.4545 


2.4750 


22 


68 


2.4750 


3.4959 


3.5171 


3.5386 


3.5604 


3.5836 


2.6050 


21 


69 


2.605O 


3.6379 


3.6510 


3.6746 


3.6985 


3.7338 


2.7474 


20 


70 


2.7474 


3.7735 


3.7980 


3.8339 


3.8503 


3.8770 


2.9042 


19 


71 


2.9042 


3.9318 


3.9600 


3.9886 


3.0178 


3.0474 


3.0776 


18 


72 


3.0776 


3.1084 


3.1397 


3.1715 


3.3040 


3.3371 


3.2708 


17 


73 


3.2708 


3.3053 


3.3402 


3.3759 


3.4123 


3.4495 


3.4874 


16 


74 


3.4874 


3.5360 


3.5655 


3.6058 


3.6470 


3.6890 


3.7320 


15 


75 


3.7320 


3.7759 


3.8208 


3.8667 


3.9136 


3.9616 


4.0107 


14 


76 


4.0107 


4.0610 


4.1135 


4.1653 


4.3193 


4.3747 


4.3314 


13 


77 


4.3314 


4.3896 


4.4494 


4.5107 


4.5736 


4.6383 


4.7046 


13 


78 


4.7046 


4.7738 


4.8430 


4.9151 


4.9894 


5.0658 


5.1445 


11 


79 


5.1445 


5.3256 


5.3092 


5.3955 


5.4845 


5.5763 


5.6712 


10 


80 


5.6713 


5.7693 


5.8708 


5.9757 


6 . 0844 


6.1970 


6.3137 


9 


81 


6.3137 


6.4348 


6.5605 


6.6911 


6.8269 


6.9683 


7.1153 


8 


82 


7.1153 


7.3687 


7.4287 


7.5957 


7.7703 


7.9530 


8.1443 


7 


83 


8.1443 


8.3449 


8.5555 


8.7768 


9.0098 


9.3553 


9.5143 


6 


84 


9.5143 


9.7881 


10.078 


10.385 


10.711 


11 059 


11.430 


5 


85 


11.430 


11.836 


12.250 


13.706 


13.196 


13 736 


14.300 


4 


86 


14.300 


14.934 


15.604 


16.349 


17.169 


18.075 


19.081 


3 


87 


19.081 


30.305 


21.470 


33.904 


34.541 


36.431 


28.636 


2 


88 


38.636 


31.341 


34.367 


38.188 


43.964 


49.103 


57.290 


1 


89 


57.390 


68.750 


85.939 


114.58 


171.88 


343.77 


00 







60' 


50- 


40' 


30' 


20' 


10' 


0' 


Deg. 



NATURAL COTANGENT. 



34 



BROWN & SHARPE MFG. CO. 



CHAPXE^R IV. 

WORM AND WORM WHEEL. 

(Fig. 8.) 




PROVIDENCE, R. I. 35 



FORMULAS. 

L = lead of worm. 

N = number of teeth in gear. 

m = threads or turns per inch in worm. 

^= diameter of worm. 
d' = diameter of hob. 
T = throat diameter. 
B = blank diameter (to sharp corners). 
C = distance between centers. 

= thickness of hob-slotting cutter. 

/= width of bands at bottom. 

d = pitch circumference of worm. 

V = width of worm thread tool at end. 
w = width of worm thread at top. 
P = diametral pitch. 
P' = circular pitch. 

s = addendum. 

/ = thickness of tooth at pitch line. 
f^ = normal thickness of tooth. 

/= clearance at bottom of tooth. 
D" = working depth of tooth. 
D" +/= whole depth of tooth. 

d = angle of thread with axis. 
If the lead is for single, double, triple, etc., thread, then 
L = P', 2 P; 3 F, etc. 



36 BROWN & SHARPE MFG. CO. 



a = 


60° to 90° 


L = 


I 

m 




P' = 


TtT 

N + 2 




D = 


_ NP' _ 

7t 


N 
P 


T = 


s... 




d = 


7t{d-2 


s) 


tan 


6 = h 

b 


r 


f^ = 


Uos S 




r^ = 


d 

- — 2 s 

2 





Practical only when width of wheel on wheel pitch circle 
is not more than ^ pitch diameter of worm. 



r'=r' + D" +/ 



B=T+2(r'-r-cos^) ^™ 
. _ -335 P' , ," 



measurement of sketch is generally 
sufficient. 



2 
d' = d+ 2f 

z; = .3iP' 

^^= .335 P' 



Note. — The notations and formulas referring: to tooth parts, given on page 5 for 
spur gears, apply to worm wheels, and are here used. 

Note. — Hob and worm should be marked, as per example : 
4 turns per i'' single .25 P'; .25 L. 
2 turns per i" double .25 P'; .50 L. 



PROVIDENCE, R. I. 



17 



TABLE OF ANGLES 

FOR GASHING WORM WHEELS. 



h-5 


^1 


OUTSIDE DIAMETER OF WORM. 


1 


li 


u 


2 


2i 


2J 


3 


3i 


4 


1.000 


1 












9042' 


70 40' 


6^21' 


5^25' 


.500 


2 






70 40' 


5° 25' 


4043' 


40 10' 


3^24' 


2052' 


2^29' 


.400 


2i 


9^42' 


7^17' 


5^51' 


4^10' 


3^39' 


3^15' 


20 40' 


2^15' 


1°57' 


.3333 


3 


70 40' 


5^50' 


4042' 


3024' 


2^58' 


2039' 


2^11' 


1^51' 


1^36' 


.2857 


31 


6^20' 


4052' 


3056' 


20 51' 


2^30' 


2^15' 


1^51' 


1^34' 


1^22' 


.2500 


4 


5^25' 


4^10' 


3° 24' 


2^29' 


2^11' 


1^57' 


1^36' 


1^22' 


1^11' 


.2222 


41 


4042' 


3^39' 


2059' 


2^11' 


1^55' 


1<^43' 


1^25' 


1^13' 


10 3' 


.2000 


5 


4^10' 


3^15' 


20 40' 


1^57' 


1^44' 


1032' 


1^16' 


1° 5' 


57' 


.1666 


6 


3^23' 


2039' 


2^10' 


1^36' 


1^25' 


1^16' 


1° 3' 


53' 


47' 


.1429 


7 


2^51' 


2^14' 


1^50' 


1^21' 


1^12' 


10 4/ 


53' 


46' 


40' 


.1250 


8 


2^29' 


1^57' 


1^36' 


1^11' 


10 3' 


57' 


47' 


40' 


35' 


.1111 


9 


2^11' 


1^43' 


1^25' 


1^ 3' 


bW 


50' 


41' 


36' 


31' 


.1000 


10 


1^57' 


1^32' 


1^16' 


57' 


50' 


45' 


37' 


32' 


28' 



38 



BROWN & SHARPE MFG. CO. 



UNDERCUT IN WORM WHEELS. 



In worm wheels of less than 30 teeth the thread of the worm 
(being 29°) interferes with the flank of the gear tooth. Such 
a wheel finished with a hob will have its teeth undercut. To 
avoid this interference two methods may be employed. ' 

Firs f Method. — Make throat diameter of wheel 

N 



T = cos^i4>^° 



+ 4^ 



or 



•937 N 



+ 4>f 



This formula increases the throat diameter, and conse- 
quently the center distance. The amount of the increase can 
be found by comparing this value of T with the one as obtained 
by formula on page ^6. To keep the original center distance, 
the outside diameter of the worm must be reduced by the 
same amount the throat diameter is increased. 

Second Method. — Without changing any of the dimensions 
we found by the formulas given on page 36, we can avoid the 
interference to be found in worm wheels of less than 30 teeth 
by simply increasing the angle of worm thread. We find the 
value of this angle by the following formula : 
Let there be 

2 y = angle of worm threaa. 

N = number of teeth in worm wheel. 



cos ;/ = i/ I _ i_ 
N 

From this formula we obtain the following values : 



N 


29 


28 


27 


26 


25 


24 


23 


22 


21 


2y 


30X 


31 


3i>^ 


32X 


32^ 


izV^ 


34K 


35 


36 


N 


19 


18 


17 


16 15 


14 


13 


12 




2 y 


Z^ 


39 


40 


41/^ 


42M 


44 >^ 


46X 


48 



20 



37 



As this latter formula involves the making of new hobs in 
many cases, on account of change of angle, we prefer to reduce 
the diameter of worm as indicated by first method, if the dis- 
tance of centers must be absolute. 



PROVIDENCE, R. 



39 



CHAPTKR V. 



SPIRAL OR SCREW GEARING. 

(Figs. 9, 10, II.) 




Fig, 9, 

RIGHT HAND SPIRAL GEARS. 

In spiral gearing the wheels have cylindrical pitch surfaces, 
but the teeth are not parallel to the axis. The line in which 
the pitch surface intersects the face of a tooth is part of a 
screw line, or helix, drawn at the pitch surface. A screw 
wheel may have one or any number of teeth. A one-toothed 
wheel corresponds to a one-threaded screw, a many-toothed 
wheel to a many-threaded screw. The axes may be placed at 
any angle. 

Consider spiral gears with : 

I. Axes parallel. 
II. Axes at right angles. 
III. Axes any angle. 



40 



BROWN & SHARPE MFG. CO. 




Fig, 10. 

LEFT HAND SPIRAL GEAR. 

Let there be : 

n"= >• number of teeth in gears •! 7 

C = center distance. 

P = diametral pitch 
P' = circular pitch. 
P" = normal diametral pitch. 
P'" = normal circular pitch. 

y = angle of axes. 

Lj = exact lead of spiral on pitch surface. 
L^ = approximate lead of spiral on pitch surface. 

T = number of teeth marked on cutter to be used when 
teeth are to be cut on milling machine. 

D = pitch diameter. 

B = blank diameter. 

" Z !- angle of teeth with axis 

/= thickness of tooth. 
s = addendum. 
D" + / = whole depth of tooth. 

Note. — Letters a and d occurring at bottom of notations refer to gears a and 3. 



I. — Axes Parallel. 
Gears of this class are called twisted gears. The angle of 
teeth with axes in both gears must be equal and the spirals 
run in opposite directions. The angles are generally chosen 
small (seldom over 20°) to avoid excessive end thrust. End 
thrust may, however, be entirely avoided by combining two 
pairs of wheels with right and left-hand obliquity. Gears of 
this class are known as Herringbone gears. They are com- 
paratively noiseless running at high speed. 



PROVIDENCE, R. I. 4I 

II. — Axes at Right Angles. 
Here we must always have : 

1. The teeth of same hand spiral ; 

2. The normal pitches equal in both gears ; and 

3. The sum of the angles of teeth with axes = 90°. 

Choosing Angle of Teeth with Axes. 

1. If in a pair of gears the ratio of the number of teeth is 
equal to the direct ratio of the diameters, /. ^., if the number of 
teeth in the two gears are to each other as their pitch diame- 
ters, then the angles of the spirals will be 45° and 45° ; for, this 
condition being fulfilled, the circular pitches of the two gears 
must be alike, which is only possible with angles of 45°. In 
such a combination either gear may be the driver, 

2. If the ratio of the diameters determined upon is larger 
or smaller than the ratio of the number of teeth, then the 
angles are : 

tan a^ = ^-^ tan ^^ = --±--^ 

In such gears the velocity ratio is measured by the number 
of teeth, and not by the diameters. 

3. Given N^, N^, and C : 

If Pa is made = P,,', then we have case '' i " and 

But if Pa is assumed, then : 

^, C7r-y2NaPJ 



and 

P ' P ' 

tan aa = -^ tan a^, = -^^ 

The gear whose P' or a is larger will ordinarily be the 
driver, on account of the greater obliquity of the teeth. 

4. Given N„, Nj, and C or D. 
See case " 7 " under III., considering ;/ = 90°. 

III. — x\xis AT ANY Angle {y). 

5. Given case " i," under II., then angles of spirals — }4 y, 
for the same reason. 

6. Analogous cases to "2" and "3," under II., may be 
worked out, when angles of axes = y, but they have been 



42 



BROWN & SHARPE MFG. CO. 



omitted, partly because the formulas are too cumbersome, and 
partly because they are to some extent covered by cases "5 " 

and "7." 

7. Given N„, N^ and C, or one of the pitch diameters. We 
find the angles by a graphic method, which for all practical 
purposes is accurate enough ; ro and v are the axes of gears 
forming angle / (see diagram, Fig. 11.) On these axes we 
lay off lines r and v representing the ratio of the number 
of teeth (velocity ratio), so that N^ : N,, : : r .? : .f z;, and 




Fig, 11. 



construct parallelogram r s v. Then, according to Mc- 
Cord,* the angles formed by the tangent s oxn the pitch con- 
tact with the axes of the gears insures the least amount of 
sliding. In bisecting angle y by tangent u and using angles 
produced in this manner we equally distribute the end thrust on 
both shafts. Both methods have their advantages ; to profit 
by both we select angles a^ and a^^ produced by tangent x, 
bisecting angle u s. 

Thus we have when angles are found and C given, 
2 C TT cos a^ cos al 



p/ji _ 



and when D^^ given 



N„ cos ajj -\- Nj, cos a„ 



p/)i 



D„ = 



D„ 7t cos a^ 

K 

7t COS ai, 



and 



* McCord, Kinematics, page 378. 



providence, r. i. 
General Formulas. 



43 






D = 



FN 

TV 



or = 



TT cos a 



B = D + 2^ 



or 



D 



pw 



p = 



or = 



p'?i 



N cos a 

P'" = P' cos a 

7t 



P" = 

~ TT 

2 

T = 

Lx = 
L = 



(Pitch of cutter.) 



or 



2X + 



lO 



N 



cos a 

N P^ 

tan a 
lo WG„ 



(6*^^ iV"^/^ 7.) 



or 



_N7^ 

Ptan a 



or 



N P' 



SG. 



tan o' cos a 
(6'^<? Note 2 and examples^ 



cos 45° 
cos' 45° 



7071 1 
.50 



No. 5 cutter for T from 


21 to 25 


Ci 5 it u .. .. 


17 to 20 


(1 „ 14 t( (I (< 


14 to 16 


a 8 " " " " 


12 to 13 



tan 45° =: 1. 000 

Note i, — Cutters of regular involute system. 
Use No. 1 cutter for T from 135 up. 
'* 2 " " " " 55 to 134 

.1 3 " " " " 35 to 54 

" 4 " " " " 26 to 34 

Note 2. — Gears used on spiral head and bed for Brown & Sharpe milling 
machine : 

W = number of teeth in gear on worm. 
Gi = " " ist " stud. 

Gi = '' " 2d " . stud. 

S = " " " screw. 

Should a spiral head of different construction be used, the formula might not 
apply. 



44 



BROWN & SHARPE MFG. CO. 



The following data is usually required in cutting spiral 
gears in a Universal Milling Machine, and it will be found 
convenient to arrange it in table form as follows : 





GEAR. 


PINION. 


"Mr» r>f Tf^f^th _____ 










Outside Diameter - - 




Angle of Teeth with Axis ----- 
Normal Circular Pitch ------ 

Pil-rVi nf r'ntfpr- ________ 








Thickness of Tooth t _-__-_ 
Whole Depth D'' + f - - ~ - - - - 




Exact Lead of Spiral 

Approximate Lead of Spiral - - - - 




Gears on Milling Machine to Cut Spiral 

Gear on Worm 

1st Gear on Stud 

2nd Gear on Stud - - - 

Gear on Screw - - - - 







If the exact lead Li can be obtained by the gears at hand, 
Li will equal Lo and we shall have from the formula 
_ 10 W G, 



(for B. & S. Milling Machine.) 



S Gi 

U _ W_G2 

10 S Gi 

Example I. 

Required the gears for cutting a spiral of 2^" lead. 

2^ I . . 

-- = - factoring, m the most simple way, we have 

i _ ^ ^ ^ _ I X 28 _ 32 X 28 W G., 
• 4"^2X2~"56x2'^56 Y64 ^ S Gi 



PROVIDENCE, R. I. 45 

Thus the gearing will be 32 T. on worm, 64 T. ist, on stud, 
28 T. 2nd on stud, and 56 T. on screw. 

Trying these gears on the Milling Machine we find that 
they cannot be used, and as we have no other regular gears 
in the ratio of 2 to i that can be used we must try, by factor- 
ing, to get such ratios for the two pairs of gears as to be able 
to use the gears at hand, bearing in mind that the combined 
ratio must be J. 

T 18 3x6 24 X 6 24 X 48 
4 ~ 72 ~ 9T~8 ~ 9 X 64 ■"" 72 X 64 

These gears are at hand and the combination can be used 
on the machine, giving the exact lead of 2^". 

Example II, 

Required the gears for cutting a spiral of 8.639" l^^d. 

8.639 = 8yVo%; reducing, by continued fractions, to a 
smaller fraction of approximately the same value, as described 
on pages 73 and 74 



639 ) 1000 ( I 
639 

361 )639( I 
361 

278 ) 361 ( 
278 



83 ) 278 ( 3 
249 



29 ) 83 ( 2 

5« 



25 ) 29 ( I 
25 



4)25(6 
24 



1)4(4 
4 



46 BROWN & SHARPE MFG. CO. 



i 1 2. 7 16 2 3 154 63 S) 

1 ? 3 Tf 2^5^ "3^ 2TT TDTJTJ- 

Selecting if as an approximation near enough for our 
purpose, and in fact as near as we are likely to find gears for, 
we have for our lead 8if . Applying the formula as in Ex- 



.mple 


1. 




W G, 






10 




S G, 






m 




216 


108 




10 




250 


125 




9 X 


12 


9 X 48 



factoring we have 

72 X 48 ,, . , 

25 X 5 100 X 5 = i"^^T^ '^^ ^^^'' required, 

these being regular gears furnished with the Milling Machine. 

Proof: 

72 X 48 X 10 _ .. -. 
= 8.640 = L9 

100 X 40 o ^ T ' 

8-639 =Li 
.001'' error in lead. 

In shops where much work is done in milling spirals it is 
desirable to have a full set of gears for the milling machine, 
from the smallest to the largest numbers of teeth that can be 
used. This makes it possible, in most cases, to get closer 
approximations than could be otherwise obtained, and often 
saves a great deal of figuring. 

When the use of continued fractions does not bring a 
close enough approximation, one method to secure a closer 
result is to add to or substract from the numerator and de- 
nominator of the fraction to be reduced, any numbers nearly 
in proportion to the given fraction, seeing that the numbers 
added or substracted are such as to make the fraction reduc- 
able to lower terms. By a little ingenuity and patience ex- 
tremely close approximations can generally be reached in 
this way. 

Take, as an illustration, the fraction in Example II. 

-6-3-9- 8639 



'TOOO" 



10 lOOOO 

Adding 9 to the numerator and 10 to the denominator, these 



PROVIDENCE, R. I. 47 

being in about the same ratio to each other as the numerator 
and denominator of the fraction, we have 

86394-9 = 8648 _ 4324 _ 47 X 92 
loooo-f 10 = looio 5005 55 X 91 

All of the gears in this case are special. 

Applying the same proof as in Example II. we find that 
this train of gears will give a lead of 8.6393+, making an 
error of .0003'' in the lead. 

No doubt a much closer approximation than even this 
could be obtained by further trial. 

Another method is to multiply both terms of the fraction 
by some number which will make one term of the fraction 
easily reducible, and adding one to or subtracting it from the 
ether term to make it possible to reduce that also. 

There is an element of uncertainty in both of these 
methods, as we never feel sure that we have obtained the 
best combination; practical work, however, rarely requires 
accuracy beyond a point that can readily be reached. 

The accompanying list of prime numbers and factors will 
be found useful in reducing and factoring fractions. 



48 



BROWN & SHARPE MFG. CO. 



PRIME NUMBERS AND FACTORS. 
1 TO lOOO. 



i 

1 




26 


2x13 


: 51 


3x17 


76 


1 

2^x19 


2 




27 


3' 


52 


2^X13 


77 


7x11 


3 




28 


2^X7 


53 




78 


2x3x13 


4 


2- 


29 




54 


2x3-^ 


79 









30 


2x3x5 


55 


5x11 


80 


2-*x5 


6 


2x3 


31 




56 


2'^x7 


81 


3^ 


7 




32 


2"' 


57 


3x19 


82 


2x41 


8 


2-3 


33 


3x11 


58 


2x29 


83 




9 


3- 


34 


2x17 


59 




84 


2^x3x7 


10 


2x5 


35 


5x7 


60 


2^ X 3 X 5 


85 


5x17 


11 




36 


2=-'x3-' 


61 




86 


2x43 


12 


2^x3 


37 




62 


2x31 


87 


3x29 


13 




38 


2x19 


63 


3^X7 


88 


2^X11 


14 


2x7 


39 


3x13 


64 


2'^ 


89 




15 


3x5 


40 


2V.5 


65 


5x13 


90 


2x3^x5 


16 


2^ 


41 




^^ 


2x3x11 


91 


7x13 


17 




42 


2x3x7 


67 




92 


2^x23 


18 


2x32 


43 




68 


2^x17 


93 


3x31 


19 




44 


2^x11 


69 


3x23 


94 


2x47 


20 


2^x5 


45 


3^X5 


70 


2x5x7 


95 


5x19 


21 


3x7 


46 


2x23 


71 




96 


2^X3 


22 


2x11 


47 




72 


2^x32 


97 




23 




48 


2^X3 


73 




98 


2x7^ 


24 


2^X3 


49 


72 


74 


2x37 


99 


3^x11 


25 


52 


50 


2x52 


75 


3x52 


100 


2^x5^ 



PROVIDENCE, R. I. 



49 



101 




131 




.161 


7x23 


191 




102 


2x3x17 


132 


2^x3x11 


162 


2x3^ 


192 


2«x3 


103 




133 


7x19 


163 




193 




104 


2^X13 


134 


2x67 


164 


2^x41 


194 


2x97 


105 


3x5x7 


135 


3^X5 


165 


3x5x11 


195 


3x5x13 


106 


2x53 


136 


2^x17 


166 


2x83 


196 


2^X7- 


107 




137 




167 




197 




108 


22x33 


138 


2 X 3 X 23 


168 


2'x3x7 


198 


2x3^x11 


109 




139 




109 


13- 


199 




110 


2x5X11 


140 


2^x5 X 7 


170 


2x5x 17 


200 


2'^X52 


111 


3x37 


141 


3x47 


171 


3^x19 


201 


3x67 


112 


2V7 


142 


2x71 


172 


2-X43 


202 


2x101 


113 




143 


11x13 


173 




203 


7x29 


114 


2x3x19 


144 


2^X3-^ 


174 


2x3x29 


204 


2-x3xl7 


115 


5x23 


145 


5x29 


175 


5^x7 


205 


5x41 


116 


2=^X29 


146 


2x73 


176 


2^X11 


206 


2x103 


117 


3^x13 


147 


3x7- 


177 


3x59 


207 


3-'x23 


118 


2x59 


148 


2^x37 


178 


2x89 


208 


2^x13 


119 


7x17 


149 




179 




209 


11x19 


120 


2"^X3X5 


150 


2 X 3 X 5^ 


180 


22x3-x5 


210 


2x3x5x7 


121 


11^ 


151 




181 




211 


1 


122 


2x61 


152 


2'^Xl9 


182 


2x7x13 


212 


2^X53 


123 


3x41 


153 


3-X17 


183 


3x61 


213 


3x71 


124 


2-^x31 


154 


2x7x11 


184 


2V23 


214 


2x107 


125 


5' 


155 


5x31 


185 


5x37 


215 


5x43 


126 


2x3^x7 


156 


2^X3X13 


186 


2x3x31 


216 


2=^X3-^ 


127 




157 




187 


11 X17 


217 


7x31 

1 


128 


2-7 


158 


2x79 


188 


2^x47 


218 


2x109 


129 


3x43 


159 


3x53 


189 


3^X7 


219 


3x73 


130 

1 


2x5x13 


160 


2-^x5 


190 


2x5x19 


220 


2V5X11 

1 



50 



BROWN & SHARPE MFG. CO. 



221 


- ■■ 

13x17 


251 




281 




311 




222 


2x3x37 


252 


2^x3^x7 


282 


2x3x47 


312 


2^x3x13 


223 




253 


11x23 


283 




313 




224 


2^X7 


254 


2X127 


284 


2^x71 


314 


2X157 


225 


3^x52 


255 


3x5x17 


285 


3x5x19 


315 


3'x5x7 


226 


2x113 


256 


2« 


286 


2x11x13 


316 


2^x79 


227 




257 




287 


7x41 


317 




228 


2^x3x19 


258 


2x3x43 


288 


2^X3=^ 


318 


2x3x53 


229 




259 


7x37 


289 


17' 


319 


11x29 


230 


2x5x23 


260 


2^x5x13 


290 


2x5x29 


320 


2«x5 


231 


3X7X11 


261 


3^x29 


291 


3x97 


321 


3x107 


232 


2=^X29 


262 


2x131 


292 


2^x73 


322 


2 X 7 X 23 


233 




263 




293 




323 


17x19 


234 


2x3^x13 


264 


2^x3x11 


294 


2x3x7' 


324 


2^X3* 


235 


5x47 


265 


5x53 


295 


5x59 


325 


5^X13 


236 


2^x59 


266 


2x7x19 


296 


2-^x37 


326 


2x163 


237 


3x79 


267 


3x89 


297 


3^X11 


327 


3x109 


238 


2x7x17 


268 


2^X67 


298 


2x149 


328 


2^X41 


239 




269 




299 


13x23 


329 


7x47 


240 


2^x3x5 


270 


2x3^x5 


300 


2^x3x5' 


330 


2X3X5X11 


241 




271 




301 


7x43 


331 




242 


2x11' 


272 


2^X17 


302 


2x151 


332 


22x83 


243 


i\' 


273 


3x7x13 


303 


3x101 


333 


3'x37 


244 


2^x61 


274 


2x137 


304 


2^X19 


334 


2x167 


245 


5X7' 


275 


5^x11 


305 


5x61 


335 


5x67 


246 


2x3x41 


276 


2^x3x23 


306 


2x3^x17 


336 


2^x3x7 


247 


13x19 


277 




307 




337 




248 


2^X31 


278 


2X139 


308 


2'x7xll 


338 


2x13' 


249 


3x83 


279 


3^x31 


309 


3X103 


339 


3x113 


250 


2x5^ 


280 


2^ X 5 X 7 


310 


2x5x31 


340 


2'x5xl7 



PROVIDENCE, R. 



51 



341 


11x31 


371 


7x53 


401 




431 




342 


2x3^X19 


372 


2^x3x31 


402 


2x3x67 


432 


2^X3'^ 


343 


V 


373 




403 


13x31 


433 




344 


2^X43 


374 


2X11X17 


404 


2-xlOl 


434 


2x7x31 


345 


3x5x23 


375 


3x5^ 


405 


3^X5 


435 


3 X 5 X 29 


346 


2x173 


376 


2-^x47 


406 


2x7x29 


436 


2^x109 


347 




377 


13x29 


407 


11X37 


437 


19x23 


348 


2^x3x29 


378 


2x3'^X7 


408 


2=^x3x17 


438 


2x3x73 


349 




879 




409 




439 




350 


2x5^x7 


380 


2-X5X19 


410 


2x5x41 


440 


2^^X5x11 


351 


3^X13 


381 


3x127 


411 


3x137 


441 


3^X7^ 


352 


2^X11 


382 


2x191 


412 


2^x103 


412 


2x13x17 


353 




383 




413 


7x59 


443 




354 


2x3x59 


384 


2'x3 


414 


2x3^x23 


444 


2^x3x37 


355 


5x71 


385 


5x7x11 


415 


5x83 


445 


5x89 


356 


2^x89 


386 


2X193 


416 


2^X13 


446 


2x223 


357 


3x7x17 


387 


3-X43 


417 


3x139 


447 


3x149 


358 


2X179 


388 


2-X97 


418 


2x11x19 


448 


2«x7 


359 




389 




419 




449 




360 


2^x3^x5 


390 


2X3X5X13 


420 


2^X3X5X7 


450 


2x3^x5' 


361 


192 


391 


17x23 


421 




451 


11X41 


362 


2x181 


392 


2'^X7- 


422 


2x211 


452 


2-X113 


363 


3x11^ 


393 


3X131 


423 


3^X47 


453 


3x151 


364 


2^x7x13 


394 


2x197 


424 


2^X53 


454 


2x227 


365 


5x73 


395 


5x79 


425 


5-X17 


455 


5x7x13 


366 


2x3x61 


396 


2^x32x11 


426 


2x3x71 


456 


2^x3x19 


367 




397 




427 


7x61 


457 




368 


2^X23 


398 


2x199 


428 


2-X107 


458 


2x229 


369 


3^x41 


399 


3x7x19 


429 


3X11X13 


459 


3'^X17 


370 


2x5x37 


400 


2*X52 


430 


2X5X43 


460 


2^x5x23 



52 



BROWN & SHARPE MFG. CO. 



461 




491 




521 




551 


19x29 


462 


2X3X7X11 


492 


2'^x3x41 


522 


2x3^x29 


552 


2^x3x23 


463 




493 


17x29 


523 




553 


7x79 


464 


2^X29 


494 


2x13x19 


524 


2^x131 


554 


2x277 


465 


3x5x31 


495 


3^x5x11 


525 


3x5^x7 


555 


3x5x37 


466 


2x233 


496 


2*X31 


526 


2x263 


556 


2^X139 


467 




497 


7x71 


527 


17x31 


557 




468 


2^x32x13 


498 


2x3x83 


528 


2*x3xll 


558 


2x3^x31 


469 


7x67 


499 




529 


232 


559 


13x43 


470 


2x5x47 


500 


22x5-^" 


530 


2x5x53 


560 


2^X5X7 


471 


3x157 


501 


3x167 


531 


3-X59 


561 


3x11x17 


472 


2^X59 


502 


2X251 


532 


2-X7X19 


562 


2x281 


473 


11X43 


503 




533 


13x41 


563 




474 


2 X 3 X 79 


504 


23x3^x7 


534 


2x3x89 


564 


2^x3x47 


475 


5^x19 


505 


5x101 


535 


5x107 


565 


5x113 


476 


2^x7x17 


506 


2 X 1 1 X 23 


536 


2'^X67 


566 


2x283 


477 


3^X53 


507 


3x13^' 


537 


3x179 


567 


3*X7 


478 


2x239 


508 


2^x127 


538 


2x269 


568 


2^X71 


479 




509 




539 


7=^X11 


569 




480 


2'5x3x5 


510 


2X3X5X17 


540 


22x3'^x5 


570 


2x3X5X19 


481 


13x37 


511 


7x 73 


541 




571 




482 


2x241 


512 


29 


542 


2x271 


572 


2^x11x13 


483 


3 X 7 X 23 


513 


3-^x19 


543 


3x181 


573 


3x191 


484 


2^x11^ 


514 


2x257 


544 


2^X17 


574 


2x7x41 


485 


5x97 


515 


5x103 


545 


5x109 


575 


5^X23 


486 


2x3^ 


516 


2-x3x43 


546 


2X3X7X13 


576 


2«X32 


487 




517 


11x47 


547 




577 




488 


2^x61 


518 


2x7x37 


548 


2^x137 


578 


2x172 


489 


3x163 


519 


3x173 


549 


3^X61 


579 


3x193 


490 


2 X 5 X 72 


520 


2^x5x13 


550 


2X5=^X11 


580 


2^x5x29 



PROVIDENCE, R. I. 



53 



581 


7x83 


611 


13x47 


641 




671 


1 

11X61 


582 


2x3x97 


612 


2^x3^x17 


642 


2x3x107 


672 


2^x3x7 


583 


11x53 


613 




643 




673 




584 


2'^X73 


614 


2x307 


644 


2^x7x23 


674 


2x337 


585 


3^x5x13 


615 


3x5x41 


645 


3 X 5 X 43 


675 


3'^X52 


586 


2x293 


616 


2^x7x11 


646 


2x17x19 


676 


2^x132 i 


587 




617 




647 




677 




588 


2-^x3x7^ 


618 


2x3x103 


648 


2->x3* 


678 


2x3x113 


589 


19x31 


619 




649 


11X59 


679 


7x97 


590 


2x5x59 


620 


2^x5x31 


650 


2X5^X13 


680 


2'x5xl7 


591 


3x197 


621 


3'^X23 


651 


3x7x31 


681 


3x227 


592 


2^X37 


622 


2x311 


652 


2-X163 


682 


2x11x31 


593 




623 


7x89 


653 




683 




594 


2x3''xll 


624 


2^x3x13 


654 


2x3x109 


684 


2-^x32x19 


595 


5x7x17 


625 


b' 


655 


5x131 


685 


5x137 


596 


2=^X149 


626 


2x313 


656 


2^X41 


686 


2x7^ 


597 


3x199 


627 


3x11x19 


657 


3-^x73 


687 


3x229 


598 


2x13x23 


62S 


2-^X157 


658 


2x7x47 


688 


2^X43 


599 




629 


17x37 


659 




689 




600 


2^X3X5^ 


630 


2X3-X5X7 


660 


2^X3X5X11 


690 


2X3X5X23' 


601 




631 




661 




691 




602 


2x7x43 


632 


2'X79 


662 


2x331 


692 


2^x173 


603 


3^x67 


633 


3x211 


663 


3x13x17 


693 


3^X7X11 


604 


2^x151 


634 


2x317 


664 


2'^X83 


694 


2x347 


605 


5xlP 


635 


5x127 


665 


5x7x19 


695 


5x139 


606 


2x3x101 


636 


2-x3x53 


666 


2X3-X37 


696 


2''X3X29 


607 




637 


7^X13 j 


667 


23 X 29 


697 


17x41 


608 


2^X19 


638 


2x11x29 


668 


2^X167 


698 


2x349 


609 


3 X 7 X 29 


639 


3^'X71 


669 


3 X 223 


699 


3 X 233 


610 


2x5x61 


640 


2^X5 


670 


2x5x67 


700 


2- X ir X 7 



54 



BROWN & SHARPE MFG. CO. 



f 

701 




731 


17x43 


761 




791 


7x113 


702 


2x3^x13 


732 


2^x3x61 


762 


2x3x127 


792 


2^x3^x11 


703 


19x37 


733 




763 


7x109 


793 


13x61 


704 


2«Xll 


734 


2x367 


764 


2-X191 


794 


2x397 


705 


3x5x47 


735 


3 X 5 X 7^ 


765 


3^x5x17 


795 


3x5x53 


706 


2x353 


736 


2^X23 


766 


2x383 


796 


2^x199 


707 


7x101 


737 


11X67 


767 


13x59 


797 




708 


2^x3x59 


738 


2x3^x41 


768 


2«x3 


798 


2X3X7X19 


709 




739 




769 




799 


17x47 


710 


2x5x71 


740 


2^x5x37 


770 


2X5X7X11 


800 


2^X5^ 


711 


3^x79 


741 


3x13x19 


771 


3x257 


801 


3^x89 


712 


2«X89 


742 


2x7x53 


772 


2^x193 


802 


2x401 


713 


23x31 


743 




773 




803 


11 X73 


714 


2X3X7X17 


744 


2'^x3x31 


774 


2x3^x43 


804 


2^x3x67 


715 


5x11x13 


745 


5x149 


775 


5^X31 


805 


5x7x23 


716 


2^X179 


746 


2x373 


776 


2^X97 


806 


2x13x31 


717 


3x239 


747 


3=^X83 


777 


3x7x37 


807 


3x269 


718 


2x359 


748 


22x11x17 


778 


2x389 


808 


2=^X101 


719 




749 


7x107 


779 


19x41 


809 




720 


2^X3-'X5 


750 


2x3x5^ 


780 


2^X3X5X13 


810 


2x3^x5 


721 


7x103 


751 




781 


11X71 


811 




722 


2x192 


752 


2^X47 


782 


2x17x23 


812 


2-X7X29 


723 


3x241 


753 


3x251 


783 


3-5x29 


813 


3x271 


724 


2^x181 


754 


2x13x29 


784 


2^X7^ 


814 


2x11x37 


725 


5-'x29 


755 


5x151 


785 


5x157 


815 


5x163 


726 


2x3xlP 


756 


2^x3^x7 


786 


2x3x131 


816 


2^x3x17 


727 




757 




787 




817 


19x43 


728 


2^x7x13 


758 


2x379 


788 


2-^x197 


818 


2x409 


729 


3« 


759 


3x11x23 


789 


3x263 


819 


3^x7x13 


730 

L 


2x5x73 


760 


2^x5x19 


790 


2x5x79 


820 


2^x5x41 



PROVIDENCE, R. I. 



55 



821 


., 


851 


23x37 


881 




911 




822 


2x3x137 


852 


2^x3x71 


882 


2x3^x72 


912 


2^x3x19 


823 




853 




883 




913 


11x83 


824 


2^X103 


854 


2x7x61 


884 


2^x13x17 


914 


2x457 


825 


3x5^x11 


855 


3^x5x19 


885 


3x5x59 


915 


3x5x61 


826 


2x7x59 


856 


2^X107 


886 


2x443 


916 


2^x229 


827 




857 




887 




917 


7x131 


828 


2^x3^x23 


858 


2x3x11x13 


888 


2^^x3x37 


918 


2x3^x17 


829 




859 




889 


7x127 


919 




830 


2x5x83 


860 


2^x5x43 


890 


2x5x89 


920 


2^X5X23 


831 


3x277 


861 


3x7x41 


891 


3^X11 


921 


3x307 


832 


2«xl3 


862 


2x431 


892 


2^x223 


922 


2x461 


833 


7^x17 


863 




893 


19x47 


923 


13x71 


834 


2x3x 139 


864 


2^X33 


894 


2x3x149 


924 


2^x3X7X11 


835 


5x167 


865 


5x173 


895 


5x179 


925 


5^x37 


836 


2^x11x19 


866 


2x433 


896 


2^X7 


926 


2x463 


837 


3^X31 


867 


3x172 


897 


3x13x23 


927 


3^x103 


838 


2x419 


868 


2^x7x31 


898 


2x449 


928 


2^X29 


839 




869 


11x79 


899 


29x31 


929 




840 


2^X3X5X7 


870 


2X3X5X29 


900 


2^x32x52 


930 


2X3X5X31 


841 


292 


871 


13x67 


901 


17x53 


931 


7^x19 


842 


2x421 


872 


2^x109 


902 


2x11x41 


932 


2^x233 


843 


3x281 


873 


3^x97 


903 


3x7x43 


933 


3x311 


844 


2^x211 


874 


2x19x23 


904 


2^X113 


934 


2x467 


845 


5x132 


875 


5^X7 


905 


5x181 


935 


5x11x17 


846 


2x3^x47 


876 


2^x3x73 


906 


2x3x151 


936 


2=^x3^x13 


847 


7xlP 


877 




907 




937 




848 


2^X53 


878 


2x439 


908 


2^x227 


938 


2x7x67 


849 


3x283 


879 


3 X 293 


909 


3^x101 


939 


3x313 


850 


2x5^x17 


880 


2^x5x11 


910 


2X5X7X13 


940 


2^x5x47 



56 



BROWN & SHARPE MFG. CO. 



941 




956 


2^x239 


971 




986 


^ 

2x17x29 


942 


2x3x157 


957 


3x11x29 


972 


2^x3^ 


987 


3x7x47 


943 


23x41 


958 


2x479 


973 


7x139 


988 


2^x13x19 


944 


2^X59 


959 


7x137 


974 


2x487 


989 


23x43 


945 


3^x5x7 


960 


2«x3x5 


975 


3x5-xl3 


990 


2x32X5X11 


946 


2x11x43 


961 


31^ 


976 


2^X61 


991 




947 




962 


2x13x37 


977 




992 


2'x31 


948 


2-x3x79 


963 


3^x107 


978 


2x3x163 


993 


3x331 


949 


13x73 


964 


2^x241 


979 


11x89 


994 


2x7x71 


950 


2X5-X19 


965 


5x193 


980 


2^X5X7=^ 


995 


5x199 


951 


3x317 


966 


2X3X7X23 


981 


3-X109 


996 


2^x3x83 


952 


2^x7x17 


967 




982 


2x491 


997 




953 




968 


2^X11-^ 


983 




998 


2x499 


954 


2x3^x53 


969 


3x17x19 


984 


2'^x3x41 


999 


3"x37 


955 


5x191 


970 


2x5x97 


985 


5x197 


1000 


2'^X5'^ 

1 



PROVIDENCE, R. I. S7 



CHAPTER VX. 

INTERNAL GEARING. 

PART A.— INTERNAL SPUR GEARiNG. 
(Figs. 12, 13, 14, 15, 16.) 

A little consideration will show that a tooth of an internal 
or annular gear is the same as the space of a spur — external 
gear. 

We prefer the epicycloidal form of tooth in this class of 
gearing to the involute form, for the reason that the difficulties 
in overcoming the interference of gear teeth in the involute 
system are considerable. Special constructions are required 
when the difference between the number of teeth in gear and 
pinion is small. 

In using the system of epicycloidal form of tooth in which 
the gear of 15 teeth has radial flanks, this difference must be 
at least 15 teeth, if the teeth have both faces and flanks. Gears 
fulfilling this condition present no difficulties. Their pitch 
diameters are found as in regular spur gears, and the inside 
diameter is equal to the pitch diameter, less twice the adden- 
dum. 

If, however, this difference is less than 15, say 6, or 2, or i, 
then we may construct the tooth outline (based on the epicy- 
cloidal system) in two different ways. 

First Method. — To explain this method better, let us sup- 
pose the case as in Fig. 12, in which the difference between 
gear and pinion is more than 15 teeth. Here the point o of 
the describing circle B (the diameter of which in the best 
practice of the present day is equal to the pitch radius of a 15 
tooth gear, of the same pitch as the gears in question) gene- 
rates the cycloid o, o', o^, o^, etc., when rolling on pitch circle 
L L of gear, forming the face of tooth ; and when rolling on 
the outside of L L the flank of the tooth. In like manner is the 
face and flank of the pinion tooth produced by B rolling out- 
side and inside of E E (pitch circle of pinion). A little study 



58 



BROWN & SHARPE MFG. CO. 



of Fig. 12 (in which the face and flank of a gear tooth are 
produced) will show the describing circle B divided into 12 




equal parts and circles laid through these points (i, 2, 3, etc.), 
concentric with L L. We now lay off on L L the distances 
o-i, 1-2, 2-3, etc., of the circumference of B, and obtain points 



PROVIDENCE, R. I. 



59 



i^, 2\ 3^, etc. [Ordinarily it is sufficient to use the chord.] It 
will now readily be seen that B in rolling on L L will success- 
ively come in contact with i', 2', 3', etc., c meanwhile moving 
to c\ r', r^, etc. (points on radii through i\ 2\ ^\ etc.), and the 
generating point o advancing to o\ o^, o^, etc., being the inter- 
sections of B with c^j c^^ r', etc., as centers and the circles laid 
through I, 2, 3, etc. Points o, o\ o^ o^, etc., connected with a 
curve give the face of the tooth ; in like manner the flank is 
obtained. 

In this manner the form of tooth is obtained, when the 
difference of teeth in gear and pinion is less than 15, with the 
exception that the diameter of describing circle B 



-y-<^-^) 



where P = diametral pitch, N and n number of teeth in gears. 
The distances of the tooth above and below the pitch line 
as well as the thickness t are determined as in regular spur 
gears by the pitch, except when the difference in gear and 
pinion is very small, where we obtain a short tooth, as in Figs. 
13 and 14. In such a case the height of tooth is arbitrary and 
only conditioned by the curve. In internal gears it is best to 
allow more clearance at bottom of tooth than in ordinary spur 
gears. 



29 TeetJi 




42 T. 



8 P. 



30 TeetJi 



Fig. 13. 




In a construction of this kind it is suggested to draw the 
tooth outline many times full size and reduce by photography. 
An equally multiplied line A B will help in reducing. 



6o 



BROWN & SHARPE MFG. CO. 




PROVIDENCE, R. I. 6l 

Second Method. — The difference between gear and pinion 
being very small, it is sometimes desirable to obtain a smooth 
action by avoiding what is termed the " friction of approach- 
ing action."* This is done, the pinion drivings by giving gear 
only flanks, Fig. 15, and the gear driving., by giving gear only 
faces, Fig. 16. In both these cases we have but one describ- 
ing circle, whose diameter is equal to the difference of the two 
pitch diameters. The construction of the curve is precisely 
the same as described under A. The describing circle has 
been divided into 24 parts simply for the sake of greater 
accuracy. 



PART B.-INTERNAL BEVEL GEARS. 

(Fig. 17.) 

The pitch surfaces of bevel gears are cones whose apexes 
are at a common point, rolling upon each other. The tooth 
forms for any given pair of bevel gears are the same as for a 
pair of spur gears (of same pitch) whose pitch radii are equal to 
the respective apex distances of the normal cones (/. f., cones 
whose elements are perpendicular upon the elements of the 
bevel gear pitch cones). (Compare Fig 19, page 67.) 

The same is true of internal bevel gears, with the modifica- 
tion that here one of the pitch cones rolls inside of the other. 
The spur gears to whose tooth forms the forms of the bevel 
gear teeth correspond, resolve themselves into internal spur 
gears (Fig. 17). The problem is now to be solved as indicated 
in the first part of this chapter. 



* McCord, Kinematics, pages 107, io8. 



62 



BROWN & SHARPE MFG. CO. 



8 P. 
Gear 40 Teeth 
JPinion 20 Teeth 




Fig. 17. 



PROVIDENCE, R. L 63 



CMAPTER VII. 

GEAR PATTERNS. 

(Fig. 18.) 

To place in bevel gears the best iron where it belongs, the 
tooth side of the pattern should always be in the nowel, no 
matter of what shape the hubs are. 

Hubs, if short, may be left solid on web ; if long they should 
be made loose. A long hub should go on a tapering arbor, to 
prevent tipping in the sand. 1° taper for draft on hubs when 
loose, and 3° when solid is considered sufficient. 

Coreprints as a rule are made separate, partly to allow the 
pattern to be turned on an arbor, partly for convenience, 
should it be desirable to use different sizes. 

Put rap- and draw-holes as near to center as possible. 
Referring to Fig. 18, make L = D for D from y^" to i>^", or 
even more, should hubs be very long. Otherwise if D is more 
than I >^" leave L= i>^^ 

Iron pattern before using should be marked, rusted and 
waxed. 

Shrinkage — For cast-iron, yi" per foot. 

For brass, yV' " 
Cast-iron gears, especially arm gears, do not always shrink 
Yi^^ per foot. In making iron patterns the following allow- 
ances have been found useful : 

Up to \2" diameter allow no shrink. 
From 12" to 18" " " Yi regular shrink. 

" 18" to 24" " '' >^ " 

" 24" to 48" " " yi " 

Above 48" " " .10" " 

for cast-iron. 



64 



BROWN & SHARPE MFG. CO. 




PROVIDENCE, R. I. 



65 



If in gears the teeth are to be cast, the tooth thickness t in 
the pattern is made smaller than called for by the pitch, to avoid 
binding of the teeth when cast. No definite rule can be given, 
as the practice varies on this point. For the different diam- 
etral pitches we would advise making / smaller by an amount 
expressed in inches, as given in the following table : 



DiAM 


Pitch. 


Amount t 
IS Smaller. 


DiAMo Pitch, 


Amount t 
IS Smaller. 




16 


.010" 


5 


.020'' 




12 


.012" 


4 


.022'' 




10 


.014" 


3 


.026'^ 




8 


.016" 


2 


.030'' 




6 


.018" 


I 


.040'' 



66 



BROWN & SHARPE MFG. CO. 



CHAPTER VIII. 



DIMENSIONS AND FORM FOR BEVEL GEAR 

CUTTERS. 

(Fig. 19.) 

The data needed to determine the form and thickness of a 
bevel gear cutter are the following : 
P = pitch. 

N= number of teeth in large gear. 
n = number of teeth in small gear. 
F = length of face of tooth, measured on pitch line. 
After having laid out a diagram of the pitch cones a d c and 
ad/, and laid off the width of face, the problem resolves itself 
into two parts : 

Part I. — Determine Proper Curve for Cutter. 
It will be remembered that in the involute system of cutters 
(the only one used for bevel gears that are cut with rotary 
cutter), a set of eight different cutters is made for each 
pitch, numbering from No. i to No. 8, and cutting from 
a rack to 12 teeth. Each number represents the form of 
a cutter suitable to cut the indicated number of teeth. For 
instance, No. 4 cutter (No. 4 curve) will cut 26 to 34 teeth. 
In order to find the curve to be used for gear and pinion 
we simply construct the normal pitch cones by erecting 
the perpendicular p q through b, Fig. 19. We now measure the 
lines b q and b p, and taking them as radii, multiplying each by 
2 and P we obtain a number of teeth for which cutters of 
proper curves may be selected. From example we have : 

Gear : b q — 9^" ; 2 X P X 9.75 = 97 T No. 2 curve. 
Pinion: b p = 3>^" ; 2 X P X 3-5 = 35 T No. 3 curve. 
The eight cutters which are made in the involute system 
for each pitch are as follows : 

No. I will cut wheels from 135 teeth to a rack. 



" 2 






(■(, 


55 






134 teeth 


" 3 






« 


35 






54 " 


" 4 






a 


26 






34 " 


" 5 






(( 


21 






25 '' 


" 6 






(( 


17 






20 " 


" 7 






(( 


14 






16 " 


'' 8 




(( 


u 


12 






13 " 



PROVIDENCE, R. L 



67 




68 BROWN & SHARPE MFG. CO. 

Part II.— Determine Thickness of Cutter. 

It is very evident that a bevel gear cutter cannot be thicker 
than the width of the space at small end of tooth ; the practice 
is to make cutter .005" thinner. Theoretically the cutting angle 
(/i) is equal to pitch angle less angle of bottom (or /i = a — fi'). 
Practically, however, better results are obtained by making 
h = a — f3 (substituting angle of top for angle of bottom), and 
in calculating the depth at small end, to add the full clearance 
(/) to the obtained working depth, giving equal amount of 
clearance at large and small end. This is done to obtain a 
tooth thinner at the top and more curved. As the small end 
of tooth determines the thickness of cutter, we shall have to 
find the tooth part values at small end. From the diagram it 
will be seen that the values at large end are to those at small 
end as their respective apex distances {a b and a I). The 
numerical values of these can be taken from the diagram and 
the quotient of the larger in the smaller is the constant where- 
with to multiply the tooth values at large end, to obtain those 
at small end. In our example we find : 

^ 7 ""—= .6s S = constant ir - -d u 

ab=c,^ "^^ For 5 P we have : 

/=.3i4i /' = .2057 

s = .2000 / = .1310 

/=.o3i4 / = :£3i4 

s +/=.23i4 / +/=.i624 

D" + /= .4314. ^' - -^310 

D"' +/=.2934 

From the foregoing it is evident that a spur gear cutter 
could not be used, since a bevel gear cutter must be thinnen 

If in gears of more than 30 teeth the faces are proportion- 
ately long, we select a cutter whose curve corresponds to the 
midway section of the tooth. The curve of the cutter is found 
by the method explained in Part I. of this Chapter. 



PROVIDENCE, R. L 69 



CHAPITER IX. 

DIRECTIONS FOR CUTTING BEVEL GEARS 
WITH ROTARY CUTTER. 

(Fig. 20.) 

In order to obtain good results, the gear blanks must be of 
the right size and form. The following sizes for each end of 
the tooth must be given the workman : 
Total depth of tooth. 
Thickness of tooth at pitch line. 
Height of tooth above pitch line. 
These sizes are obtained as explained in Chapter VIII. 
The workman must further know the cutting angle (see 
formula on page 13 and compare Chapter VIII.), and be pro- 
vided with the proper tools with which to measure teeth, etc. 
In cutting a gear on a universal milling machine the opera- 
tions and adjustments of the machine are as follows : 

1. Set spiral bed to zero line. 

2. Set cutter central with spiral head spindle. 

3. Set spiral head to the proper cutting angle. 

4. Set the index on head for the number of teeth to be cut, 
leaving the sector on the straight or numbered row of holes, 
and set the pointer (or in some machines the dial) on cross-feed 
screw of millmg machine to zero line. 

5. As a matter of precaution, mark the depth to be cut for 
large and small end of tooth on their respective places. 

6. Cut two or three teeth in blank to conform with these 
marks in depth. The teeth will now be too thick on both their 
pitch circles. 

7. Set the cutter off the center by moving the saddle to or 
from the frame of the machine by means of the cross-feed 
screw, measuring the advance on dial of same. The saddle 
must not be moved further than what to good judgment 



70 



BROWN & SHARPE MFG. CO. 




Eig. 20. 



PROVIDENCE, R. I. Jl 

appears as not excessive ; at the same time bearing in mind 
that an equal amount of stock is to be taken off each side of 
tooth. 

8. Rotate the gear in the opposite direction from which the 
saddle is moved off the center, and trim the sides of teeth (A) 
(Fig. 20.) 

9. Then move the saddle the same distance on the opposite 
side of center and rotate the gear an equal amount in the 
opposite direction and trim the other sides of teeth (C). 

10. If the teeth are still too thick at large end E, move the 
saddle further off the center and repeat the operation, bearing 
in mind that the gear must be rotated and the saddle moved 
an equal amount each way from their respective zero settings. 

It is generally necessary to file the sides of teeth above the 
pitch line more or less on the small ends of teeth, as indicated 
by dotted lines F F. This applies to pinions of less than 30 
teeth. 

For gears of coarser pitch than 5 diametral it is best to 
make one cut around before attempting to obtain the tooth 
thickness. 

The formulas for obtaining the dimensions and angles of 
gear blanks are given in Chapter III. 



72 



BROWN & SHARPE MFG. CO. 



CHAPTER X. 

THE INDEXING OF ANY WHOLE OR FRAC- 
TIONAL NUMBER. 

(Fig. 21 ) 




Change Gear 

Fig. 21, 



In indexing on a machine the question simply is : How- 
many divisions of the machine index have to be advanced to 
advance a unit division of the number required. To which 
is the 

divisions of machine index 
answer = 



number to be indexed 



Suppose the number of divisions in index wheel of machine 
to be 2i6. 



Example I. — Index 72. 
Answer: 216 



72 



= 3 (3 turns of v/orm). 



PROVIDENCE, R. I. 73 

Example II. — Index 123. 

— =i + -93 
123 123 

If now we should put on worm shaft a change gear having 
123 teeth, give the worm shaft, Fig. 21, one turn, and in addi- 
tion thereto advance 93 teeth of the change gear (to give the 
fractional turn), we would have indexed correctly one unit of 
the given number, and so solved the problem. Should we not 
have change gear 123 we may try those on hand. The ques- 
tion then is : How many teeth (x) of the gear on hand (for 
instance 82) must we advance to obtain a result equal to the 
one when advancing 93 teeth of the 123 tooth gear? We have : 

-^ = -- where x = ^^ 
123 82 

Example III. — Index 365, change gear 147. 

— = -i where ;^ = 87 — -^ 
365 147 365 

Here 147 is the change gear on hand. In indexing for a unit 

of 365 we advanceS^teeth of our 147 tooth gear. It is evident 

that in so doing we advance too fast and will have indexed 

three teeth of our change gear too many when the circle is 

completed. To avoid having this error show in its total amount 

between the last and the first division, we can distribute the 

error by dropping one tooth at a time at three even intervals. 

Example IV. — Index igo. 

216 _ .26 

7^ — ^ "•" — ^ Change gear on hand 88 T 

— = -^ where j = 12 + 

190 88 190 

To distribute the error in this case we advance one additional 
tcoth ot a time of the change gear at eight even intervals. 

Example V. — Index 117.3913. 

216 _ 986087 



117.3913 1173913 

This example is in nowise different from the preceding 
ones, except that the fraction is expressed in large numbers. 
This fraction we can reduce to lower approximate values, 
which for practical purposes are accurate enough. This is 
done by the method of continued fractions. [For an explana- 



74 BROWN & SHARPE MFG. CO. 

tion of this method we refer to our " Practical Treatise on 

Gearing."] 

986087 





II739I3 


986087) 


II739I3 (I 




986087 




187826) 986087 (5 




939130 




46957) 187826 (3 




140871 




46955) 46957 (^ 




46955 




2) 46955 (23477 




46954 




1)2(2 




2 




. 0" 




986087 _ , 




"V39M , ^ I 




5+1 




3 + 1 




i + i 




23477 + 1 
2 


I 


5 c=s I 23477 2 



^1=1 ^ = 5 d = 16 21 493033 986087 
a^ = 1 If^ = 6 d^ = ig 25 586944 1173913 



^ ■ the 



Note. — Find the first two fractions by reduction = - and — , — - , 

•^ II I + I 6 

5 

others are then found by the rule j <^ ^ + ^ — " 



The fraction W is a good approximation; putting therefore 
a change gear of 25 teeth on worm shaft, we advance (beside 
the one full turn) 21 teeth to index our unit. 

Of course, in using any but the correct fraction we have an 
error every time we index a division ; so that when indexed 
around the whole circle, we have multiplied this error by the 
number of divisions. 

In the present example this error is evidently equal to the 
difference between the correct and the approximate fraction 
used. Reducing both common fractions to decimal fractions 
we have : 

-5 — = .84000006 

1173913 

21 

— =.84000000 

.00000006 = error in each division. 



PROVIDENCE, R. I. 75 

.00000006 X 1 17.3913 = .00000704348 total error in complete 
circle. This error is expressed in parts of a unit division. (To 
find this error expressed in inches, multiply it by the distance 
between two divisions, measured on the circle.) In this case 
the approximate fraction being smaller than the correct one, 
in indexing the whole circle we fall short .00000704348 of a 
division. 

Example VI.- 




15708 

983) 1309(1 
983 

326) 983 (3 
978 
5) 326 (65 
30 
26 
25 

1)5(5 
5 
o 



983 
1309 



+ 1 

3+1 

65 + 1 
5 

3 65 



I 3 196 983 



I 4 261 1309 

In using the approximation |^f ^ the error for each division 
(found as above) will be .000002927, for the whole circle 
.0000460. In this case, the approximation being larger than 
the correct fraction, we overreach the circle by the error. 



/6 



BROWN & SHARPE MFG. CO. 



CHAPTER XI. 

THE GEARING OF LATHES FOR SCREW 
CUTTING. 

(Figs. 22, 23.) 

The problem of cutting a screw on a lathe resolves itself into 
connecting the lathe spindle with the lead screw by a train of 
gears in such a manner that the carriage (which is actuated by 




Simple a earing. 

Fig. 22. 



PROVIDENCE, R. I. 



77 



the lead screw) advances just one inch, or some definite dis- 
tance, while the lathe spindle makes a number of revolutions 
equal to the number of threads to be cut per inch. 

The lead screw has, with the exception of a very few cases, 
always a single thread, and to advance the carriage one inch it 
therefore makes a- number of revolutions equal to its number 




Compound Gearing- 

Fig, 23. 



of threads per inch. Should the lead screw have double 
thread, it will, to accomplish the same result, make a number 
of revolutions equal to half its number of threads per inch. It 
follows that we must know in the first place the number of 
threads per inch on lead screw. 



78 BROWN & SHARPE MFG. CO. 

It ought to be clearly understood that one or more inter- 
mediate gears, which simply transmit the motion received from 
one gear to another, in no wise alter the ultimate ratio of a 
train of gearing. An even number of intermediate gears 
simply change the direction of rotation, an odd number do not 
alter it. 

The gearing of a lathe to solve a problem in screw cutting 
can be accomplished by 

A. Sim. pie gearing. 

B. Compound gearing. 

Referring to the diagrams, Figs. 22 and 23, we have in Fig. 
22 a case of simple, and in Fig. 23 a case of compound gear- 
ing. 

In simple gearing the motion from gear E is transmitted 
either directly to gear Ron lead screw or through the interme- 
diate F. In compound gearing the motion of E is transmitted 
through two gears (G and H) keyed together, revolving on the 
same stud ;?, by which we can change the velocity ratio of the 
motion while transmitting it from E to R. With these four 
variables E, G, H, R, we are enabled to have a wider range of 
changes than in simple gearing. 

B and C, being intermediate gears, are not to be considered. 
If, as is generally the case, gear A equals gear D, we disregard 
them both, simply remembering that gear E (being fast on 
same shaft with D) makes as many revolutions as the spindle. 
Sometimes gear D is twice as large as gear A, then, still con- 
sidering gear E as making as many revolutions as the spindle, 
we deal with the lead screw as having twice as many threads 
per inch as it measures. 



SIMPLE GEARING. 

Let there be : the number of teeth in the different gears 
expressed by their respective letters, as per Fig. 22, and 

s = threads per inch to be cut, 
L ■— threads per inch on lead screw ; then 

I. ^ ^ 5: 

L E 



PROVIDENCE, R. I. 79 

If now one of the two gears E and R is selected, the other 
will be : 

L s 

2. The two gears may be found by making 

-p ""f T f where/ may be any number. 

3. The above holds good when a fractional thread is to be 
cut, but if the fraction is expressed in large numbers, as, for 
instance, s = 2.833 (2TTHfV)» we first reduce this fraction (y^o^) to 
lower approximate values by the process of continued fraction 
(see pages 73 and 74). 

833) ICOO (I 

833 
167) 833 (4 

668 

165) 167 (I 
165 
2) 165 (82 
16 

"^ 

1)2(2 
2 
o 

I 4 I 82 2 



_^ _£ _5_ 414 833 

156 497 1000 



± = .S^^ (nearly) and s = 2± 
6 6 

If in this case L = 4, and we select E = 48, then, since 
R = ^ R = 34 



COMPOUND GEARING. 

4. In a lathe geared compound for cutting a screw the 
product of the drivers (E and H, Fig. 23) multiplied by the num- 
ber of threads per inch to be cut must equal the product of the 
driven (G and R) multiplied by the number of threads on lead 
screw. This is expressed by 

E.H.^=G.R.Lor ^— i^- = i 



So BROWN & SHARPE MFG. CO. 

If three of the gears E, H, G, R have been selected, the 
fourth one would be either 

^ GR L 

H = ^^ or 

E s 

n E H ^ 

G = or 



R 



S=: 



R L 
E H^ 

GL 
RG L 



VL.E.H/ 



E H 

If a fractional thread is to be cut, as under " 3," we reduce 
the fraction to lower approximate values. 

Example.— Gear for 5.2327 threads per inch, lead screw is 

6 threads. 

2^27 
.2327 = -^J- 

lOOOO 
— 2327) lOOCO (4 

9308 
692) 2327 (3 

2076 

251) 692 (2 
502 
190) 251 (I 

190 

"61) 190 (3 

183 

7) 61 (8 

5)7(1 
5_ 

2j 5 (2 
4 

i) 2 (2. 
2 

o 
43213 8 I 2 2_ 

13 7 10 37 306 343 992 2327 



4 13 30 43 159 ^3^5 1474 4263 loooo 

— = .2327 (nearly) and 5.2327 = 5 — 
43 "43 

Selecting E = 43, H = 52, R = 50, and 

^ E . H . j- , ^ 43 . 1^2 . K^^ 

G = we have G = ^ — ^ ii-5 = ^g. 

R . L 50 . 6 '^^ 



PROVIDENCE, R. I. 8l 

5. The examples so far given all deal with single thread. 
The pitch of a screw is the distance from center of one thread to 
the center of the next. The lead of a screw is the advance for 
each complete revolution. In a single thread screw the pitch 
is equal to the lead, while in a double thread screw the pitch 
is equal to one-half the lead ; in a triple thread screw equal to 
one-third the lead, etc. 

If we have to gear a lathe for a many-threaded screw 
(double, triple, quadruple, etc.), we simply ascertain the lead, 
and deal with the lead as we would with the pitch in a single 
thread screw, /. ^., we divide one inch by it, to obtain the num- 
ber of threads for which we have to gear our lathe. 

Example. — Gear for double thread screw, lead = .4654. 
Number of threads per inch to be geared for is : 

—1-= -1-=: 2.1487 
Lead -4654 

Lead screw is four threads per inch. 
As in previous examples, we reduce the fraction .l4S']=-^Jf-^-^^-Q 
to lower approximate values by the process of continued frac- 
tion. 

From the different values received in the usual way we 
select : 

\l = .1487 (nearly) and 2.1487 = 2^ 

We have therefore : 

L = 4 

(E=74 
Selecting ^ G = 30 

( H = 40 
G . L 30 . 4 

Note, — In using any but the original fraction we commit an error. This error 
can be found by reducing the approximate fraction used to a decimal fraction, and 
comparing it with the original fraction. In the above example the original fraction is 

.1487 and 
H = . 14864 



Error = .00006 inch in lead. 



In cutting a multiple screw, after having cut one 
thread, the question arises how to move the thread tool the 
correct amount for cuttimx the next thread. 



82 



BROWN & SHARPE MFG. CO. 



In cutting double, triple, etc., threads, if in simple or com- 
pound gearing the number of teeth in gear E is divisible by 
2, 3, etc., we so divide the teeth ; then leaving the carriage 
at rest we bring gear E out of mesh and move it forward one 
division, whereby the spindle will assume the correct position. 

When E is not divisible we find how many turns (V) of 
gear R are made to each full turn of the spindle. Dividing 
this number by 2 for double, by 3 for triple thread, etc., we 
advance R so many turns and fractions of a turn, being careful 
to leave the spindle at rest. 

For compound gearing : 

G.R 

When the gear D is twice as large as the gear A (as ex- 
plained in fifth paragraph, page yS.) the formula would be 

2 G. R. 

If in simple gearing both E and R are not divisible, one 
remedy would be to gear the lathe compound ; or the face- 
plate may be accurately divided in two, three or more slots, 
and all that is then necessary is to move the dog from one slot 
to another, the carriage remaining stationary. 



